The next item of business is a member's business name in the name of Ross Thompson on the impact of north-east businesses of the hike in non-domestic rates. A debate will be concluded without any questions being put. Would those members who wish to speak in the debate please press their request-to-speak buttons now? I call on Ross Thompson to open the debate. Thank you very much, Presiding Officer, and I'm delighted to bring this member's debate to the chamber this afternoon. I thank other members for the cross-party support in enabling this to come forward. North-east businesses right across the region have been desperately crying out for help with their business rates revaluation, warning of the devastating impact to their own business and also their community if nothing was done to help. I launched a petition calling for support, which accumulated more than 2,000 signatures in the space of only 48 hours. Businesses told me time and time again that they risk going to the wall with jobs lost if something wasn't done. I'm proud to speak for each and every one of those businesses this afternoon. The Scottish Government, sadly, is simply trying to pass the buck. From the outset, we were told that the Government would not intervene as this was a matter for the independent assessor and for local councils to deal with. To give credit to Derek Mackay, he did come to Aberdeen to hear for himself the very real concerns of local businesses. However, the Scottish Government's attempt to dress up extra funding for all Scottish councils as income that could be used to mitigate against business rates rises fooled no-one in the north-east business community. Aside from the fact that every council in Scotland received a top-up and only Aberdeen City and Aberdeenshire were expected to spend that income on rates relief, Mr Mackay and the SNP declined to mention that the overall budgets for all local authorities were still being slashed. A cut is still a cut. Now, only after the criticism of this Government hit the national press and Mr Mackay faced calls to make his statement to Parliament have we seen action. To coin an Angus Robertson phrase, we saw an embarrassing and screeching U-turn from the cabinet secretary, which followed hot on the heels of the reversal of the ill-thought-out plan to raise raid north-east council budgets to send the income to the central build. Now, a 12.5 per cent cap will be welcomed by the hospitality sector and will make a difference to the hefty bills levied on office premises in the north-east. However, based on the Scottish Government's own figures, it appears that there will be still about 8,000 businesses in the region that will not benefit from those changes. Furthermore, the cap will only be in place for one year, when businesses say that they need certainty over at least three years. The Scottish Chamber of Commerce hit the nail on the head by saying that the Scottish Government's so-called bespoke solution for the north-east will not deal with the root of the problem. It is nothing more than a sticking plaster. It is crystal clear to everyone in the north-east that a fundamental re-examination of the whole business rate system is essential, and I hope that the recommendations of the Barthley review will provide much-needed relief and support for businesses in the north-east. The cabinet secretary is right on one thing, however. The north-east is facing exceptional circumstances, but what is he going to do about it? Both Aberdeen City and Aberdeenshire councils have identified sums of money to help alleviate the problem in the face of savage cuts to their own budgets from the Government. Mr Thomson rightly identifies that the local authorities have the ability under the Community Empowerment Act to put in their own rates relief schemes. Why is it that Tory councillors in Aberdeenshire and Aberdeen have voted against such schemes? Has he told businesses across Scotland when he has been going out and about that that is what his Tory colleagues have done in both councils? He spends more time in Edinburgh than he does in the north-east, because if he allowed me to make progress, I was about to say that I am pleased with the role that I have played in securing £3 million in local funding for rates relief in Aberdeen City. If Derek Mackay was to be true to his word that the north-east faces exceptional circumstances and that he is willing to work with local councils, I am sure that he will match that funding and accept that Aberdeen City has invited him to meet with them, to discuss that programme, and I am sure that he will be more than happy to do so. Match funding such a scheme could allow councils to support the businesses and sectors that are currently not being helped by the Scottish Government cap. Mr Mackay has the money to do it, because he can use some of the £320 million extra from the UK Government to put in place funding for councils to support businesses affected by rates rises. The UK Government has announced a £300 million relief fund for local authorities for business rates relief in England. I tell the Scottish Government today that business rates rises and increases are not something that you can ignore and just hope that it is going to go away. More must be done. On top of that, Mr Mackay says that local councils keep every penny of business rates that they raise, yet it has recently come to our attention that Aberdeen Shire Council will raise £23 million more than was allocated by the Scottish Government in its local government settlement. Will the Scottish Government provide a cast iron assurance—since it likes those—that those missing millions will stay in the local area and will not be swallowed up here in Edinburgh? I have heard that missing millions catchphrase in the local press. I am really surprised that you, as a sitting councillor, do not understand how budgets work in councils. I have not been a councillor and I understand that it is over a five-year period that money is allocated to councils that is not based on the tax take of one year. As a sitting councillor, I can tell Julie Martin that, last year, just as this year, we had to set a one-year budget. The Government is giving us no clarity on our future. We have to set our budget looking into the future and not knowing what that future is going to be. I also know that we keep getting told that every single penny of non-domestic rates we raise, we keep and we don't. That is the evidence that was presented to Aberdeenshire Council and that is the evidence that I am providing to Julie Martin this afternoon. It is our duty to do all that we can to help businesses, particularly those in the northeast who have endured difficult times due to the downturn in oil and gas. Unfortunately, SNP tax policy on non-domestic rates, the large business supplement, council tax, income tax and stamp duty are hindering economic growth and hammering our region. It is time to do more. I remind the chamber that I am the PLO for the Cabinet Secretary for Communities, Social Security and Equalities, and I am also a sitting councillor. I would like to thank Cross Thompson for bringing this motion forward and allowing us to discuss this vitally important issue in the chamber today. I am sure that, like many others here, our mailboxes have somewhat dominated in recent times. I was initially contacted by the Belvedere hotel in Stonehaven, who raised concern at the increases that are affecting businesses in the town and hospitality businesses in particular, and met personally with them and many others. The Belvedere hotel faced a 106.6 per cent increase in the rates, but they were by no means alone. One guest house was facing a 250 per cent increase in the rates substantially above the 37 per cent average increase nationally. It had compiled a list of all businesses and outlining those affected. Of 355 businesses that it had listed in Stonehaven, 87 were affected by the new revaluation. Of those, a number were hospitality businesses, but there were, of course, other problem areas. I was contacted about another case by a landlord on behalf of the tenant regarding building for agricultural use. The use was queried by the assessor's office, who has yet to make a final decision, but in the meantime the tenant is being asked to pay £130,000 worth of rates, as well as an extra £13,200 surcharge for the writ that is there to receive demanding the payment. Then there are the retail premises. I recently met the owners of the hardware store in Lawrence Kirk, who told me about the 100 per cent increase seen in their business rates. That is a business that has made no alterations or grown in size and which, over the past few years, has seen its footfall and takings a drop. The decline in footfall for this business actually started when the banks pulled out of town. The people from the surrounding areas who would go to Lawrence Kirk for the bank would frequent the shop—a slight side-point, but one that highlights the impact that such closures have on our rural communities. Had the rates not increased here, the business would in fact now be eligible for the extension of the small business bonus, even though they may well be if they are successful on appeal. However, it is not just the actual increases that are the problem. Those individual cases highlight the fundamental issues that there are with the current rates system. It is how the rates are determined, how those valuations are reached, the frequency and how often that is done, and the appeals process itself. Although businesses can appeal, the problems lie, and they have six months in which to lodge that appeal, but, as the Belvedere pointed out to me, they could potentially wait another two years for that appeal to be heard, during which time the higher rates have to be paid enough to force some businesses under. What will be of fundamental importance going forward is the Berkley review that I expected in the summer, which I sincerely hope will address some of the concerns raised by local businesses in my constituency, to me and across the region. The chambers of commerce, the Federation of Small Business, among many others, have called for changes to that system to address some of those problems raised. Asking for a reduction in bureaucracy, making the process more transparent, more frequent valuations and a system that is more adaptable to our changing economy have fundamental importance in the north-east. They also call us for a system that incentivises business creation and investment. However, although all of us eagerly anticipate the outcome of the Berkley review in the summer, we also need to recognise the good news in all of that. The tens of thousands of businesses who have been lifted out of business rates altogether by the measures taken by this Government—I am sorry, I am just in my final closing seconds, but I will happily discuss it with member after. Of the businesses that I mentioned in Stonehaven earlier, 54 have benefited from the new threshold lifting them out of rates altogether, while the remainder of that 355 pay no rates at all. There are the extra relief packages introduced by Aberdeenshire Council over and above the support that was announced by the Government a few weeks ago, which has helped many of the businesses who contacted me initially. I would have hoped that there would have been more support for that from Mr Thomson, but, given his party's actions, or should I say, lack of action in both Aberdeenshire and Aberdeenshire, I suppose that I cannot be surprised at some of his comments. However, one thing is for sure that, while the Tories will continue to cart from the sidelines, the SNP will be the ones who are continuing to work for our communities and for all the businesses in those communities. Liam Kerr, to be followed by Lewis MacDonald. I thank Ross Thomson for bringing in the issue to the chamber. It is without doubt one of the biggest issues that we face in the north-east right now. Today, I want to highlight the impact on a sector that has had little mention in the chamber, but which, according to Gary Walton of BusinessRatesAdvice.com, has been particularly hard hit by our nurseries. Nurseries in the north-east are, on average, facing an increase of 50 per cent in their rates. For you to know that there is not one nursery in Aberdeenshire that is going to not be assisted by the new business rates relief scheme that has been put forward by the administration of the Aberdeenshire Council. I thank Gillian Martin for her incorrect intervention, because, as I am going to go on and talk about, I have a very particular example. Perhaps she should have given me longer to develop the argument. Nurseries in the north-east are, on average, facing an increase of 50 per cent in their rates, with the highest being a staggering 177 per cent. More than one faces the prospect of closing their doors completely in the face of such increases. All this in the context of a Scottish Government aiming to increase free childcare from 600 to 1200 hours per year. Councils across the country already face a dilemma of where to put these children, as there are not enough nursery spaces, a problem that will only be exacerbated if nurseries close. Last week, in response to a huge number of emails from many nurseries and their clients on this issue, I visited the Croft Nursery in Stunhaven. It and its two other sites in Stunhaven and Chapleton are excellent establishments, exemplars of good practice, good facilities and innovative thinking to build the best environment for children, built over 25 years by entrepreneur Linda Peary. Linda's nursery's face rate increases of 54 per cent, 28 per cent and a staggering 82 per cent respectively. She faces a choice, hike prices to parents, many of whom cannot pay, as north-east unemployment rises and pay freezes by making more sense for parents, especially women, to stay off work and provide the care themselves, or she closes. Linda began to cry, as she was telling me this. This is real. This is real people, real families, real businesses and real jobs. In Lawrence Kirk's three weeks ago, I sat with a number of local businesses who had got together following no response to their letters to the SNP and they had asked me to come down and see them. In a town that has lost key facilities that draw people in, such as its petrol station and its bank. In the context of an oil sector downturn and reduced footfall across the board, I was particularly concerned by Marin's hardware limited, which I believe is the one that Mary Evans visited this week. A small retail hardware shop on the high street whose rates are going up by 100 per cent. At best, they may have to cut the hours of their staff. At worst, they close their doors completely. There is yet another property empty on our high streets, redundancies in the local economy and another loss to Lawrence Kirk. Thanks to Scottish Conservative Pressure and the P&J's excellent campaign, we secured a 12.5 per cent cap for the hospitality sector and for office space. However, as I have pointed out in this chamber frequently, in the north-east, over 10,000 businesses still face increase in their bills and get no relief at all from the Government. At general questions today, Derek Mackay trotted out his line that the councils might help if Linda begs them. Aberdeenshire Council, as we all know, is one of the lowest-funded in Scotland and faces further swinging cuts. The Government will say to appeal, but, as Linda told me just yesterday, this is another cost. Retaining an expert to do this on their behalf and they are expected to pay until an appeal can be heard, two years time at least. Those businesses say, I see you and the north-east Conservatives trying hard, I see the P&J campaigning to try and save us. I have not heard anything from the SNP constituency MSPs. They are right for on this issue as so many others, the people of the north-east, when needing their MSPs to stand up to the Scottish Government are met with stony silence. Will Linda have written to the First Minister again, she did not get a reply last time, so I asked the question, will the First Minister bother to respond this time? We shall see. Presiding Officer, I commend Ross Thompson's motion for the dangers that it highlights and the terrible consequences for the north-east if this Government does nothing. I fully congratulate and thank the P&J for its effective campaign. Derek Mackay said earlier today that this Government will support people through revaluation and ask that we trust them to support businesses. Now is the time to deliver and help our businesses before it is too late. I congratulate Ross Thompson on bringing today's debate. I think that all members who represent the north-east should know just how tough times have been for the city region of Aberdeen in the past two years. For many people who are running businesses there, the rates revaluation has seemed like the final straw. Part of the problem is that the new values that are placed on their premises are not based on the rental value now—two years into the downturn. Instead, they are based on a tone date of 1 April 2015, when the impact of the falling oil prices had not yet worked its way through the supply chain or the regional economy. All the trend statistics over the last two years confirm that Aberdeen and Aberdeenshire have been hardest hit by the downturn, whether that is the increase in unemployment or the fall in property prices and values. None of that is reflected in the rates revaluation that we have seen. That is not about appeals by individual businesses. I have had ministers respond to questions that I have raised by standing up in Parliament and saying that any business can appeal, and therefore it is not an issue for the Government. That simply insults the intelligence of ratepayers. If everyone you know is in the same situation and that you know that the whole of your sector across your city region has been overvalued, then you know that the issue is not the individual assessment—it is the approach overall. Ministers have recognised that that is true, if only for some sectors, but the fact that they have recognised that at all is welcome. The problem, though, is that by capping two sectors without addressing the root problem with assessments in Grampian, that has only exacerbated the problem for everybody else. Ministers have not yet explained why they accept that it is a sectoral problem in office, accommodation and hotels, but they do not recognise the same need for action in nurseries and childcare. The Bridges' Nursery in West Hill and Bridgedon is facing an increase in rateable value for their premises equivalent to more than the cost of an additional member of staff. The same, I am sure, is true for Linda Piddy, whose business in Stonehaven and Chavelton has been mentioned by Liam Kerr. Ministers accept that the downturn in the price of oil and gas has hurt the Aberdeen city regions economy, but they have done nothing with the selective caps to mitigate the impact of the revaluation on companies in the oil and gas supply chain. Northsea compactors are specialist engineers who design and manufacture heavy-duty waste compactors for offshore platforms and rigs. They are facing 100 per cent increase in rates. Graham Dawson, body and paint shop limited, faces a revaluation that has taken them over the threshold so that they no longer benefit from any rates relief as a small business, yet the amount of business that they do has not increased at this difficult time. Graham and Linda Dawson, like others, fear that the increase that they face could put them out of business. Ministers have implicitly accepted that there is a structural problem with the revaluation in the grantian area, given regional economic circumstances, yet they still respond to questions as if appeals by individual companies were the only solution. Precision oil tools may well appeal against their 60 per cent increase in rates. Their previous rating bill was actually calculated in March 2015, only weeks before the nominal date of the current revaluation, and yet that has gone up by 60 per cent. Even if they decide to appeal, they know that they will have to pay the new rates in full in the meantime, and they are deeply concerned by what that may mean in the next few weeks and months for their loyal and hard-working staff. Clearly, the Scottish Government has to act to sort out this problem. It is their responsibility. They need to provide local councils with the resources to let them take the action that is so urgently required, not just in hospitality, not just in office accommodation but in nurseries and childcare, in the oil and gas supply chain and across the economy. Only if all businesses are treated fairly in these tough times will all business owners believe that their voices are being heard, and I urge ministers to act now before businesses have to lay off workers to pay their bills or go to the wall altogether. Before I start, I refer the chamber to my register of interest, especially those concerning the business sector. It is my experience in this sector from which I speak today, and I am grateful to my colleague Ross Thompson for bringing forward his motion. It is vital that businesses feel that they are represented in Parliament, though it is clear that SNP members regard business and business people with disdain and as a block in their never-ending quest for separation. The SNP do not understand business, and Alex Salmond Gawn has given up all pretence of trying. They need to remember that it is businesses that power our economy, but employ people and create jobs and, importantly, pay the taxes that are necessary to support our public services. So when businesses say they are in trouble, the Scottish Government needs to listen, and it is no wonder that I have been contacted by so many businesses in my area surrounding the crippling hike of business rates. From nurseries to restaurants, from oil companies to those in renewables, none are falling for the sticking plaster of a 12.5 per cent cap, which, of course, will last only a year. The Scottish Retail Consortium has gone as far as to say that this sticking plaster only adds further complexity to an already ffiendishly complicated system, whilst others have labelled the plaster as a half-baked back-of-a-fag-packet fix. In my constituency, a local nursery still faces the maximum 12.5 per cent hike for this year. Certainly an improvement on the 65 per cent rise that they were facing a month ago before I flagged this at First Minister's Questions, but this still does not address the main problem, namely that the nursery has no choice but to pass these extra charges on, meaning that parents can afford less nursery time, disproportionately preventing mothers returning to work. So the SNP are not only strangling the north-east, but simultaneously destroying the balance sheet of Scotland. With no money for further relief, empty property rates will lead to capital value being destroyed in a return to the days of window and roof taxes. Derek Mackay is still trying to plaster over his mistakes. At the Finance Committee, he said, it is true to say that every local authority area keeps every penny of non-domestic rates. He then went on to add, every council area will still keep every penny that it raises. This position was repeated by Gillian Martin and Paul Wheelhouse at an Inverrory Business Association only a matter of weeks ago. However, this is simply not true. The total money raised by Aberdeenshire Cancer will be £116 million. The total money that will be available to the council will be £93 million. I ask the finance secretary where our missing £23 million is. Will you keep your word and make sure that rates raised locally are spent locally? Perhaps we will find out following the Conservative amendment, which forces the SNP to find the backbone to represent their constituents and write to the finance secretary. It is clear that only the Scottish Conservatives will stand up for the north-east, while the SNP is prepared to cripple Scotland's economy at any cost. First of all, I congratulate Ross Thomson for bringing this debate to us. It is an extremely important debate and I acknowledge Ross's work in this area, particularly over the months to bring it to everybody's attention. I do get a little bit set up with only the Conservatives' stand-up for the north-east. Ross Thomson used in his opening remarks the phrase that we should do all we can, and I agree entirely with him. I was about to say that I agree with everything and every word that has just been said from the conference-server benches until the last speaker. Words are important but they are easier than actions. As I said, I gently say to the Conservative colleagues from the north-east particularly, last week we had an opportunity to do something about this when we looked at the local government finance order. What the local finance order does is it distributes the money from the Scottish Government to the councils. We all know that Aberdein City Council receives the worst handout from the Scottish Government, and Aberdein City Council is the third worst and has done for many years under this administration. I have long given up on asking SNP members from the north-east to stand up for the north-east, but they just do not do it. We have three members, city members, who are members of the Government. Of course, I would not ask them to resign from the Government because they would have to do that if they voted against the local government order, but that restriction does not appeal to other north-east MSPs—applied to north-east MSPs—from the Conservative benches. I do not want to sound patronising because I am not trying to be, but there are five new Conservative MSPs from the north-east, and they have obviously listened to the advice from their finance spokesperson who told them that they were going to abstain on the finance order, but that is not the way to stand up for the interests of the north-east. I would urge next year and the following year not to listen to that sort of advice, or at least to try behind the scenes to persuade your finance spokesperson to let Conservative MSPs from the north-east really represent the interests of the people of the north-east and press that button against the SNP Government when you believe that what they are doing is wrong. This is a Parliament. It is not just about standing up and saying the right thing that you are doing, but it is actually using the vote that people send you here to use in the right circumstances. I have never been afraid to do that myself, and I am still here. I have disappeared and come back again, but it has not done me any harm, I can tell you that, despite difficulties within my own party when I did it, but we need to have a little bit more independence of mind and freedom to do the right thing. As I say, my comments tonight are directed to my Conservative colleagues, and there are bigger issues facing the nation over the next couple of years, which I hope we can all speak with the same voice. We also need to speak up for the north-east, not just speak up for the north-east, use that button, use those votes, but also give credit where it is due. I have given credit to Ross Thompson and the efforts of the north-east Conservative colleagues, but you are not the only ones. When we get change, it is much better done across the party, rather than being entirely partisan. As I say, I have given up on the SNP, at least I have a hope of the Conservatives, thank you. Thank you. Before I call Bill Bowman, there is a great deal of interest in this debate, and might I suggest that members may wish to extend the debate under rule 8, 14.3. Can I invite Ross Thompson to move such a motion to extend the debate? I move. Thank you. If members are in agreement, we are in agreement. We will extend the debate by up to 30 minutes, as is necessary. Can I call on Bill Bowman to be followed by Julian Martin. Thank you, Presiding Officer. Usually, in my script, as Deputy Presiding Officer, it's nice to be before you today. As a bit of a new boy, I thank Mike Rumbles for those words of advice, which I will take note of. I was just, as an accountant, as you do, trying to do some arithmetic. I think that there's more conservative in the chamber now than everybody else combined. I may be wrong, but I think that we are taking an interest in our region, maybe not quite the way that you would like us to, but we are trying. I join my colleagues in thanking Ross Thompson for bringing this debate forward. Like fellow colleagues, I am aware of a number of businesses in the region who have been hit by significant increases in their business rates. Businesses that lengthen breadth of Scotland are genuinely concerned about the critical effect that those increases are going to have on the future of their business and, ultimately, their workforce. Many examples have been cited in the course of this afternoon's debate, and I would want to add a further one, that of Dundee Football Club. The club will be presented with a bill for business rates that is 63 per cent more than what it has paid up until now. An increase that represents the biggest percentage rise in business rates to affect any football club in the Scottish premiership. That is a prime example of a business that will not benefit from any relief or the a cap announced by the finance secretary last month. Truth of the matter, however, is that it will be the fans who will be the ones bearing the brunt of the increase. As we all know, it is the market that determines what people pay for a product, and even if it is only an extra pound or two at the turnstiles or an increase in the price for a match day or season ticket, it is a cost that both fans and clubs could well do without. Sadly, as we have heard timing again today, Dundee Football Club does not stand alone when it comes to such increases. In Dundee, 20 per cent of rateable properties will see an increase in their bill. There are over 1,100 businesses that are going to have to, as was put to me by one constituent who got in touch, just to find the money from somewhere to pay the additional cost. As has been expressed already today, my fear is that this will result in job losses and, in some cases, some businesses, particularly small businesses, questioning whether they will be able to afford to survive. That is desperate news for our economy and the economy of our local communities. There has been a steady flow of businesses coming out against those increases and calling on the Government to do something. The chief executive of the Scottish Licence Trade Association said that the rates could be the last straw for many pubs, while the owner of Tay Park House in Dundee said that the rate rises were a threat to smaller businesses. While I completely accept that the assessors are independent of Government, it is within the Scottish Government's powers to offer support to those affected most. Instead, what we got was dither and denial before the announcement of some relief measures, which, of course, I welcome and will undoubtedly help some businesses affected. But why were Scottish businesses left waiting in the lurch so long? The head of policy for the Scottish Retail Consortium, as we have heard already, described Mr Mackay's measures as yet another sticking plaster. While the Federation of Small Businesses said that the furory associated with this year's revaluation shows why the system is long overdue for reform, a remark that chimes with those of ratings experts from Rhydon's Commercial Property Services in Dundee who described the rates as all over the place. In conclusion, it is clear that this debacle has left many businesses worried about their future and with no confidence in the valuation system. It is time for the Scottish Government to listen to such concerns and act to address them. An additional £160 million was pledged to local authorities by the finance secretary ahead of the budget last month. That additional local funding was part of a budget that was decided to grow the economy and fund our public services in the north-east of Scotland. Aberdeinshire Council got one of the largest shares of that £160 million—the £8 million—and, with the first council to pledge that it would use the bulk of that funding for business support. Not only that, the Scottish Government will reduce the rate poundage and the core tax that applies to the rateable value of business properties by 3.7 per cent. That is in addition to the small business bonus scheme, which has already saved businesses more than £1 billion across Scotland and will be expanding from April to lift 100,000 businesses out of rates completely. In towns like Ellen and Tarrif in Aberdeinshire East, which is my constituency, that is the difference between vibrant high streets and empty high streets. In Inverdory, which is one of the areas in Scotland that has a particularly high property and rental values for a very long time, many high street businesses have rateable values that are above that threshold. Aberdeinshire Council has decided that businesses with a rateable value of between £15,000 and £18,000 will have their relief from the Scottish Government's increase to 50 per cent, which is an extra 25 per cent. The scheme also includes approximately £208,000 for economic development support and about £105,000 for an extension of the empty property relief. It is all about what we mentioned about the Scotland-wide cap rates in hospitality businesses and particular provisions for the north-east for office space. In Aberdeinshire, that is assisting 150 hospitality premises and 159 offices. The Aberdeinshire Council administration, which is a coalition between SNP, Labour, Greens and Independence, committed funding to give 50 per cent reduction in rates increases to business, with a rateable value of £120,000 or under. I would like to finish that point and then I will take you, Mr Kerr. There are only 217 businesses with a rateable value of over £120,000 in Aberdeinshire, and not one nursery has a rateable value of above that threshold. They will be eligible for the rates relief scheme, so I find it a little bit disingenuous that nurses are getting mentioned some more. I take the points that nurses need some assistance, but they are getting that assistance from Aberdeinshire Council. I must say that I was slightly surprised at the bravado of Mr Thomson's motion, but I thank him from the bottom of my heart for allowing us to have this member's debate, because it gives me the chance to point out that his Conservative colleagues on the Aberdeinshire Council not only voted against such a relief system, but in their alternative budget they did not include any mention of our rates relief scheme for any businesses at all. Does she agree with Linda Piddy of the Croft Nurseries in Stonehaven and Chableton when she says that the increase in rates affecting her nursery business in Aberdeinshire is unfair and extortionate? That is why the Barclay review is taking place, and I think that all those businesses that are finding that they are getting the hiking rates and they are maybe working with the assessors and they are not getting any change of valuation around their circumstances. They should absolutely be contacting the Government and putting their views forward to the Barclay report, because that is the kind of thing that it is going to address. I have just taken an intervention, so I would like to get ahead. The missing millions thing that you have all been talking about—I would just like to read out from Spice about how local finance works. Non-domestic rates income is currently the single largest source of revenue under the control of the Scottish Government. The administration of business rates and rates release scheme is a matter for each local authority, but that is what I want to say. Each council having collected its taxes reports the non-domestic rates collected to the Scottish Government to be included in the central pool. The amount to be redistributed from the pool is known as the distributable amount in the set by the Scottish Government. It is based on a forecast of non-domestic rates income and prior year adjustments. That is how it works. How can you say, right, that there is going to be this year a collection of non-domestic rates, so therefore this is what we have planned for a year before? That just doesn't make sense. Yes? Ross Thompson, but we have a time for time. Presiding Officer, if Julie Martin had looked through the rest of that Spice briefing, I am sure, you would see that the total NDR take and the distributable amount, which is NDRI, for Aberdeen, is going to be 6 per cent down in real terms for 1718 and Aberdeenshire is going to be down 4.1 per cent. Julie Martin, on time to wind up if we made this person. The non-domestic rates take, it goes into the pool and then it is going to be provided to the local authorities in future years based on the spend that they are asking for. It has always been a bit of a treat of fans of slapstick to behold the Tories and Aberdeenshire speaking in the press, but some kind of fantasy that money has been withheld from my area. The vast majority of businesses in Aberdeenshire are being lifted out of rates. I take Ellen as an example. I am going ahead. I want to say the last thing for us to sit down, Presiding Officer. Council tax payers deserve a little bit more respect, and I think that the Tories are giving them. They cannot expect that people will not notice it, but despite all their wailing in the press, they tried to vote down a budget that included a business rate rate scheme worth over £3 million, in the same way that they could not put forward a council budget that specified taking over £8 million away from education, and they stand up in this place and criticise school staff shortages. People's heads do not zip up their back. We can all see behind the bluster. Peter Chapman, to be followed by Jackie Baillie. I would also like to thank my colleague Ross Thompson for bringing this debate to the chamber today. Back in budget season, which seems so long ago now with all that has happened since, I asked our local SNP MSPs whether they would stand up for businesses against the rate rises. Needless to say, they did no such thing. Instead, Gillian Martin stood up at a business breakfast in Inverroory and offered no support to owners facing hikes and business rates, putting the very future of their business at risk. She even had the gall to suggest that it was something to blame on Westminster for not supporting the oil and gas industry enough. Ladies and gentlemen, that was absolutely disgraceful. Of course, Gillian Martin. The point that I was making at that business breakfast is that we should be supporting all businesses. I was very concerned about some of the rhetoric that was going along about small businesses—this was not from yourself, Mr Chapman—but someone said that small businesses should be ashamed to not be paying rates. I take real history with that because there is a lot of struggling small businesses that populate our high streets. With regard to Westminster, what I was saying is that you have got to get a bit of a cheat, because the Westminster is not supporting the oil industry by having any kind of tax concessions or loan guarantees that have been asked for by the oil and gas UK repeatedly. Peter Chapman. I reiterate—we have heard it again—that it is an absolute disgrace. £2.3 billion in tax cuts is the most beneficial tax regime anywhere in the world, and if we had an independent Scotland, there would be none of that support for the oil and gas industry. Anyway, even Gillian Martin's predecessor, Alex Salmond MP, did not think that that line would work. He set out our finance security on the straight and narrow, and he dropped a wee hint to Nicola and Chums—sorry, I should not say the SNP cabinet—in a video blog that he was not happy about, quote, legitimate concerns being ignored by the SNP Government. That is what he said. It is a shame that the once great double act of Salmond and Sturgeon has been so reduced that the former First Minister has to communicate with the SNP Government through social media and the local paper. I guess that Nicola Sturgeon and Derek Mackay were too feared to take his calls, and so the finance secretary was humbled by the conservative opposition when he was dragged before this chamber after the grown-ups had told them where to get off. He was just stised by parties across the chamber for not going far enough, but he had kept the party leader happy—sorry, the former party leader—but there remain problems on the ground. Aberdeenshire councillors were not playing the game of being grateful for the crumbs of Mr Mackay's table. They rounded in him for having the gall to say that he was boosting their funding as he gave with one hand and took away much more with the other. In other words, a second-rate deal from a second-rate finance secretary. This little story, ladies and gentlemen, takes us up to the present day, except all is not said and done in this matter. To return to the finance secretary's role in all this, he still has some explaining to do. I suspect that we will receive no answer on this, because his boss, the MP for Gordon, has not asked him the question yet, but I do hope that he will see the value in explaining his thinking. Back when he spoke to Aberdeenshire council on this matter, he made it abundantly clear that any money raised locally would be for the local authority to spend. However, Aberdeenshire council, as we have already heard, will collect £116 million in business rates and will only receive £93 million back. Despite what everyone else on that side of the chamber says, there is £23 million that has vanished. Maybe he was distracted by setting up a fundraising website for a second divisive independence referendum, when he should have been looking after taxpayers' money, or else it is another example of the SNP's inability to understand deficits. Maybe it is down the back of Mr Mackay's sofa for a rainy day, but I think that people of Aberdeenshire deserve an answer from the cabinet secretary, because he promised that money raised locally would stay local, he needs to explain why he is now breaking that promises. Businesses in my part of the world expect politicians to do what they say. When rate rises hit local businesses such as the spotty bag shop in Banff, and the tufted duck hotel or even international powerhouses like SCORE, they need to know that that increase will go to help their local community. If the finance secretary does not believe in local taxes being spent locally, he should just say so. Anything less is an insult to the business owners of the north-east, who will be left paying for the SNP's next vanity project. It is of critical importance to the business community, particularly in the north-east of Scotland, where the impact is more significant than in any other area of Scotland. Four minutes is not a great deal of time, so I will not test your patience. I am sure that members will forgive me if I simply cut to the chase. The Scottish Government has delayed the revaluation of business rates, which is why I believe that the increase for so many businesses is so high. As Lewis MacDonald rightly said, the current revaluation is based on property values before the decline in oil prices when the economy was performing better than it is today. Businesses tell us that they want revaluations more often, so there are not dramatic increases, and I note that the rest of the UK is moving to three-year revaluations, and the Scottish Government should do at least the same. We have heard stories of the very difficult impact that those increases will have. I have a hotel in my area that was going to be subject to something like a 200 per cent rise in their rates bill. That would have had a direct impact on jobs. You can imagine how hard that is in a small rural area where they are a significant local employer. Then there is the hydro project. It used to be zero-rated, then it had a rates bill of 36,000, now they expect their rates bill to soar to something like 160,000. That is quite simply huge, unforeseen and unplanned. The Scottish Government says that they share a desire to develop renewables. Let me say that as gently as I can. You will not find many people pursuing hydro projects in the future when margins are already tight if the rates increases that they are facing. The Scottish Government, under pressure that I acknowledge, has provided some relief that is welcome, but it is only in place for one year. What happens next year is anybody's guess. 31 per cent of businesses will see a net increase in their bill. With 100,000 small businesses now exempt from rates because of the small business bonus, 56 per cent of non-small businesses will see their rates rise. As we warned earlier in the month, key sectors such as universities will see a 27.1 per cent increase, hospitals and 11.6 per cent increase, or when there are significant public sector funding challenges. I believe that the SNP Government can and should do more. We have heard lots of talk today. I want to see practical action. Today, I want to set out to the chamber exactly what Scottish Labour would do. I would encourage the SNP to take our idea. Steal it. I would be happy to give it to you. Implement it. Let's see that practical action, because as a result of the recent UK budget statement and changes to business rates provision in England, Scotland will receive some £36 million over the next three years in additional money. Labour's proposal is to take all of that money and apply it to business rates relief, additional to what is being provided by the Scottish Government. We want to see this money given directly to local authorities because they know best what is needed on the ground for their local businesses. We would base the allocation to local authorities on the basis of those areas that are worst affected, which would see the north-east receiving the most and other authorities receiving amounts proportionate to the scale of their problem. Aberdeen, who has seen an increase in business rate bills of 62 per cent, Edinburgh, an increase of 38 per cent, Glasgow of 27 per cent and at the other end of the scale, Dundee at 20 per cent would all receive support. Derek Mackay and the SNP cannot hoover up the extra cash coming to Scotland. It needs to go straight to the communities that need it most. Local councils should decide where the relief goes based on the needs of their local economy. They have the power to deliver, but they do not have the necessary resources to help businesses. Scottish Labour's proposal would provide practical help at a local level. I commend it to the minister because this must not be a nationalist cash grab, because Labour has a plan for a fair deal for business. I reiterate my thanks to Ross Thompson for bringing today's debate forward and reminding members of my declaration of members' interests as a councillor on Murray Council. I ask to speak in today's debate because it has rightly focused on the north-east of Scotland and a small chunk of the Murray council area, namely Bucky, comes into the north-east region of the Scottish Parliament, but clearly the Murray council and Murray businesses face similar struggles to those throughout the north-east. As the only member representing Murray who has sat through this debate, it is important to put on record some of the issues that we have locally as well. A spice briefing released today confirms that out of 4,540 business properties in Murray, 1,590 will not be included in any of the SNP's rates relief proposals. That is 43 per cent of Murray businesses who will continue to face crippling increases in their business rates with no support at all from the SNP Government. Compare that to the 77 per cent of businesses in Fife who will benefit from the proposals put forward, and we have to ask why the north-east of Scotland and why areas such as Murray are being made to suffer by this SNP Government. I would like to put on record, as I have in the past, the welcome move by the SNP Government after a continued campaign by those of us on the benches and the press and journal newspaper to cap the increases for the hotel industry at 12.5 per cent. That was welcomed by the Clooneybank hotel in Forrest, by the beach bar and the stockfield hotels in Lossymouth and many other businesses. However, as the chief executive officer of the Murray Chamber of Commerce said, although the short-term help is welcomed, we cannot take our foot off the gas and we must work with partners for a long-term solution. There was some welcome in Murray to the proposals put forward by the SNP, but it was nothing more, as Alexander Burnett said, than a sticking plaster approach, and we have to look for more. I also briefly want to mention a lot of the discussion that we have heard from SNP members about the support of SNP councillors for their local businesses. I would like to ask the SNP members, and perhaps the minister will indulge me with his response. However, would he agree with me that if SNP councillors, such as those in Murray, are going to put forward a proposal that they expect to get backed by other elected members, they should have some detail in the proposal, they should have some costings in the proposal and they should have a budget? We were faced in Murray council to support a proposal by the SNP that was so vacuous that it had none of that information. That was the same SNP group in Murray council who were able to do some research and find £1,000 for town centre clock face upgrades. They found a priority for town centre clocks, but not for our businesses and I think that that is shameful for them. Finally, I want to mention appeals and it has been mentioned by several speakers. There are 43 per cent of businesses in Murray who will not benefit from the rates put forward by the SNP Government and they will want to appeal to the Grampian Assessor. We have heard from members on all benches that appeals can take up to two years and that is something that is unacceptable as they continue to pay these higher prices during the two years. What reassurance can the Scottish Government and the minister give us that appeals will be fast tracked to ensure that businesses are not out of pocket while they claim their appeals? What we need to be done is businesses in Murray who face huge increases and those across Scotland need the help of this Parliament and their Government. We need assistance and a new system that provides clarity, fairness and reassurances that our businesses are looking for and expect and not one-year stick-and-plaster approaches that have some short-term benefit but not long-term gain. I welcome this opportunity to respond on behalf of the Scottish Government on the important issue of the outcome of the business rates revaluation for businesses in the north-east of Scotland, which comes into force on 1 April 2017. The revaluation, which is undertaken by independent assessors appointed by local government, is the first to take place since 2010 and broadly spans the recovery from the last economic downturn. Our overarching priority has been to maintain a competitive rates regime and we have engaged directly with a range of businesses and their representative organisations, including those in the north-east, to respond to their concerns. The Government acted early to deliver a competitive business rates package alongside an improved funding settlement for local government. Recognising that the revaluation could lead to increased costs, the draft budget for 2017-18, published in December, set out a highly competitive package, notably reducing the rates poundage by 3.7 per cent, expanding the small business bonus scheme so that it will lift 100,000 properties out of rates completely and limiting the application of the large business supplement so that 8,000 fewer properties will pay it. Together, those measures were already set to reduce the overall rates burden by £155 million and ensure that 70 per cent of businesses will pay less than or the same as this year, with more than half paying nothing at all. I will take Mr Rumbles. Would the minister not have been better to respond to the debate and the issues that are raised by individuals rather than reading out a pre-typed script? It would be very helpful if he could do that. Fully intend to go through a lot of the stuff that folk have gone through today, but I am laying out what was in the original draft budget. It is important that we lay the ground for that. We also recognise the particular challenges that are faced by the north-east economy, with central and local government having a role to play in ensuring that local issues are addressed. Having continued to engage with and to listen to businesses since the draft budget, we have acted to provide further support where it is most needed in light of the re-evaluation. Accordingly, we will ensure that, across Scotland, businesses, restaurants, pubs, hotels and cafes will see their bills increased by no more than 12.5 per cent on April 1. Additional support is injected into the north-east economy to recognise the impact of the oil and gas downturn, and the reliefs for renewable sector are expanded. Those measures amount to an additional £44.6 million of support, which offers proportionally double the total relief package that is announced by the UK Government on 8 March. I will take Mr MacDonald yet. I am very grateful. Can the minister explain today why the Government made the judgment that there should be additional support for the cost of office accommodation in Aberdeen, but no additional support for businesses in the oil and gas supply chain, which have been hit by substantial increases in their rateable advantage? Mr MacDonald is probably well aware that Mr Mackay and others have spoken to stakeholders in the north-east of Scotland, including Aberdeen and Grampian. That is why our overall package for the north-east will mean that around 13,700 properties will pay no rates at all as a result of the small business bonus scheme. It is why we are capping the increases at 12.5 per cent for hotels, pubs, restaurants and cafes, which will benefit almost 1,100 properties, and will provide an additional £5.7 million. Capping bill increases also at 12.5 per cent for offices in Aberdeen in Aberdeenshire, will benefit around a further 1,100 properties and provide additional relief worth an estimated £4.8 million. Mr MacDonald also stated in his speech earlier that it will take years for an appeal to be heard. I just clarify to all members here today that businesses who appeal can request that their appeal is heard within 70 days under the legislation. Let's not go around telling businesses that it will take years to appeal when it can be done within 70 days. Let's look at some of the other things that have been said during the course of this debate. It was said by both Ross Thompson and Alexander Burnett that Aberdeen Council is not getting to retain its business rates. All business rates that are collected are, in accounting terms, paid into a central pot and then returned to local authorities. Prior to 2011-12, it was done on the basis of population. Since 1 April 2011, each council now retains every penny of business rates that it collects. As a result of that, there is no need for redistribution. What they have done today, Presiding Officer, is showing that they have a fundamental misunderstanding of local government finance to suggest that councils do not retain all the business rates that they collect. However, there seems to be quite a few fundamental misunderstandings. We have a situation where Mr Thompson, who is also a councillor in Aberdeen City, had the opportunity to vote for a local rates relief package in the city. He did not even turn up for the budget to support an SNP proposal that outlined £4 million of support. I will take you in a minute, Mr Thompson, which would have seen rate rises capped at 12.5 per cent for the retail sector and manufacturing sectors, and the rates capped at 3 per cent for the hospitality sector. Maybe Mr Thompson can tell the businesses that he has been round about why he did not agree with the measures that were put forward by the SNP group in Aberdeen City. Ross Thompson. First of all, yes, you are right. I was at the budget meeting. I was here voting against his Government's awful budget. It was giving an awful deal to Aberdeen. Also, if you look at the minutes of previous committee meetings, Kevin Stewart did the same thing at the 11-12 budget 2. In relation to the £4 million, the SNP put forward a proposal for £4 million, yes, to help the hospitality sector. What the city council has put on the table is £3 million to devise a scheme in the chamber of commerce for all other sectors who are not getting help, and with the match funding, we would take that to £6 million, far better than the £4 million that the SNP is trying to put on the table. Kevin Stewart. As I pointed out, it was not just the hospitality sector that the proposals were put forward for, it was the retail and manufacturing sectors too, and he should have paid more attention to that. Aberdeenshire councillors, Tory councillors, voted down a similar package in Aberdeenshire. I am sorry, I am closing Ms Bailey. We have seen lots and lots of noise from the Conservatives. We heard from Mr Thomson about all the things that he has been doing, but there is absolutely no correspondence from Mr Thomson to the Government on that issue. At the very least, one would have thought that that would have been the case. We are in a situation where we take cognisance and have responded to the north-east of Scotland situation, but the Cabinet Secretary for Finance and the Constitution has already given an undertaking in evidence to the local government and communities committee that he will engage fully on any increased scrutiny of decisions on non-domestic rates with the Parliament and its committees. I agree with Mr Mackay that the time to do that is following the external review led by Ken Barkley, which is currently engaging businesses to explore how right rates might better reflect economic conditions and support investment. I think that he should maybe listen to this, Ms Bailey, instead of shouting from the sidelines, because we will look to support that investment in growth and the Government will respond swiftly when the review concludes this summer. I ask for your clarity, and if you will invite the minister to confirm to the chamber, he made it very clear to every member who raised an issue about the appeals process that businesses are able to appeal and ask for that appeal to be heard after up to 70 days. Can you ask if the minister will confirm to Parliament that his Scottish National Party Government has resourced the local assessors to deal with every appeal within the next 70 days? That is a message that I will put out to my constituents in Murray, and I just want to make sure that the Scottish Government is prepared for that. That might be a point for the minister, but it is not a point of order. I thank you for this debate, and we adjourn this meeting until 2.30.


Welcome to module 46 of point set topology part 1. Last time we stated and proved Tyknoff's theorem, of course we assume Alexander's Sub-Base theorem. So in order to complete the proof of Tyknoff's theorem, we should now prove Alexander's Sub-Base theorem. We have already made some set theoretic preparation for that also last time. So let us start with the proof of Alexander's Sub-Base theorem, which I restate here that X be at a level space and S be a subspace of its topology. X is compact if and only if every cover of X by sub-family of F admits a finite sub-cover. The necessity of this condition for compactness is obvious because X is compact every open cover has to admit a finite sub-cover. After all members of S are open, so some sub-family of S covers means there is an open cover. So that must admit a finite sub-cover, that condition is necessary. The crux of the matter is that we have to prove a converse. So we expected that this proof will be sufficiently complicated. So we have to be ready for that. So fix a sub-base S for X and let us have notation that B is the base generated by S, namely elements of B are those which are finite intersection of members of S. So in what follows, I shall use the word cover to mean a cover for X, so that that much shortness of notation is short time, words, okay, a little bit of small word set. What we have is, let us just recall what is the meaning of whatever you are talking about. Every cover U of open subsets means the content is at tau means U is a sub-family of tau means they are open subsets admits a finite sub-cover. This is the compactness, right. The second statement is every cover F contained inside B, that means what? Only members of this particular B are allowed. This is a smaller family than tau after that admits a finite sub-cover. The third one is even shorter. Every cover epsilon contained in the sub-base admits a finite sub-cover. Obviously A implies B implies C, okay. We have also seen that B implies A, okay. Now what we want to prove is C implies B. So this is the gist of Alexander sub-base theorem. So indeed what we shall do is C implies the contrapositive of B, namely if F is a family of B which has no finite sub-cover then it is not a cover which is the same thing as if it is a cover then it is a finite sub-cover. So it is in this form I am going to prove C implies B, okay. So this is all just to clarify the ground situation. Now the plan of attack is as for fix a family contained inside B which admits no finite sub-cover. Construct F's of family, another one like this, namely these are all now families of families, okay.  Finally we want to show that it is not a cover for X, okay. So F contained inside H and H has no finite sub-cover, okay. So look at all those H which also have this property no finite sub-cover but they are larger than F. So that is my family of, so families of tau here. The first claim is that you seem to show that under the set theoretic inclusion the partial order set theta which is containment. So it is a partial order has at least one maximal element. From one arbitrary F we want to have something which is maximal, okay with respect to this property that it has no finite sub-cover, okay. We have to assume that is one F then only this theta will be non-empty we are unsure. Then we have the maximal element for that we have to apply John's lemma which means we have to prove something there, okay this is a plan of that. Second step will be take such a maximal element H in theta. Now put this epsilon equal to H intersection H that means those members of H which are in the sub-base, those members of H which are in the sub-base, the sub-base is fixed. Okay, so that is a smaller family than H, right? Okay, this does not admit any finite sub-cover because H does not admit any finite sub-cover. So how can epsilon admit finite sub-cover? If this does not admit finite sub-cover from the hypothesis in seed follows that this sub-family epsilon is not a cover for X, okay. So up till here we have arrived by using John's lemma and our assumption that there is a F such that which is now does not admit finite sub-cover, okay. Now second claim is look at the union of all members of this H and so that is clearly, so this is a claim, sorry, look at all the members of H that is not a cover, that is what we know, it does not admit finite sub-cover but now take the entire set, we would show that this is contained inside unions of members of this epsilon. So what we have concluded for epsilon, epsilon does not cover X, so this will also not cover X. So that will complete the proof. If this union of all elements in H is contained in the union of all elements in this epsilon and epsilon does not cover X, so H is also not a cover of X. If H is not a cover of X, remember H was having some maximal property, in particular F is contained inside H. So if H does not cover, F does not cover, okay. So that will complete the proof. So we have to prove two steps here. In the first claim, we have to prove the John's lemma, whatever hypothesis needed, that is the first part and then you have to prove this claim, okay. Once we prove this claim, claim two is over once that the proof will be completed. So it remains to prove claim one and claim two. So let us do it one by one. Look at claim one, let me just recall, okay. This theta has a maximal element is what we have to show. For that, what is the ingredient that you have to put inside the John's lemma? Take any chain inside theta, we must show that that chain has an upper bound. This is what I will show. This part is very easy. As usual, quite often let phi be a chain in theta. Recall what the meaning of chain? Chain is a totally ordered subset of theta under the usual inclusion here, okay. Consider the family G, which in the union of all members of this chain, obviously under the inclusion that will be an upper bound for this one. But don't hurry, that must be an element of theta. Then only it will be justified. The larger set of all subsets, it is an upper bound, fine, okay. So clearly G contains F because each member of this member of this chain, they are members of theta. All the members of theta contain F, okay. So it is a subfamily of B also because each member of theta is also subfamily of B. So if you take unions of all these members, you take each, all the members are inside B. So that is not a problem. Moreover, suppose finitely many members of G say G1, G2, Gn cover X. This is the last part which I have to show that it is inside theta. That no finite family covers X, that is what you have to show, right. If not, suppose there are G1, G2, Gn belonging to G, which cover X. Remember what was G? G is just union of members of this chain. So G1 will be inside say some lambda 1, G2 will be lambda 2, G3 lambda 3 and so on. But these are all one contained in the other. You take the maximum of these. When you have finitely many of them, you have the maximum. That will contain all the G1, G2, Gn, okay. But that is a member of this theta. So it would cover, so it is a contradiction, okay. So I repeat all these Gis are in one single element G prime of B. But this G prime is a family which belongs to theta by definition. It has no finite sub cover for X. So this is observed because we assumed that G admits a finite sub cover. So G does not admit finite sub cover. That qualifies it to be a member of theta. Therefore, every chain in theta has an upper bound in theta. Once you have satisfied this property, John Slema tells you that theta has a maximal element, okay. So first claim is done. Second claim, take an element which is a maximal element for theta. Any maximal element, fix that, that is call it as H. Clearly, F contains H. So our aim is to show that F does not cover X. So you showed H does not cover X. In fact, what we will show is the following. Namely, the claim which says that claim is even stronger than what we need. Namely, union of all members of H is actually contained inside union of all members of is epsilon, okay. Remember this epsilon comes from the sub basis, the sub family of sub basis, okay. And we know that this does not cover. So this is what we have to prove, okay, which is stronger than just showing that F does not cover, okay. Let us prove this one now. Take a member here, U inside H. By the very definition, U is a member of the base. A member of the base means it is intersection of finitely many elements S1, S2, SN from the sub basis. We claim that one of the SIs is actually inside H, okay. See start with some U inside B, okay. We want to show that that U is concerned in the union of these things, okay. Now what we end up is saying that one of these SIs is actually inside H, okay. If this is not the case, let us say it is not inside H. So this is another sub claim. You may say claim 3, but claim 2 is not yet done. It is part of claim 2. So suppose one of these S1, S2, SN is not inside H. If this is not the case, then consider family HI, which is H union singletonite. Add this one more member. You get another family. Add H1, add S1. You get one family, H1. Add S2, you get H2. Add S3, you get S3 and so on. So you get H1, H2, H3, which are all larger than H. If I have put one extra element and that I am assuming is not inside H. Each of them is a sub-family of B, okay. So every element inside this curly S is inside B also and contains H, which of course contains F. But by maximality of H, these HI are not members of theta. You see, they are larger than because each H is a maximal element. This can happen only if each of them, each of them admits a finite sub-cover. There are members in this family. Just that union of them is a finite cover for H, okay. So what are those members? If you pick up only members for a match, that is not going to cover. So each time you have to put SI also. But just SI may not cover. Along with some members here, finitely many, this SI will cover. That is the meaning of that, okay. So I get, for each I, see UI1, UI2, UI, UI, PI, 1 less than K less than 2, up to PI, UI1, UI2, etc. These are the members of HI, union, actually they are members of H, union 1 SI. All these UIKs are members of H. And what are they? They are, this union is whole of X, okay. SI has to be there. SI is not there. That will be give you H is a cover, finite cover. That is the assumption that H is the, H doesn't have a finite cover. Now look at all these UIKs from H1, H2, H3 and so on. Along with, instead of S1, S2, SN, okay, you just take the intersection, okay. And now the Kruskal's matter is that this will be a finite sub cover for X. Clearly it is finite. It is a finite sub cover for X. But these are all members of H only now. These are the members of H. You, you starting with the, I remember H. Why this is the cover for X? Take any point, if they are inside UIKs, any fix one PI here, fix one I, if they are UIKs, that is fine. Otherwise, they will be in the corresponding SI. Each time they are here, that's okay. But if it's not here, none of, none of these UIKs contain a point X, then this point must be inside SI for every I, which is the same thing as saying that point is inside U. Therefore, this cover, okay. So this violates the fact that H is belongs to theta because it is a finite sub cover. So what we have proved is a substatement here that one of the SI's, let us call it as S1 is already in H, okay. So that just means that this S1 is an element of this epsilon, right. See, S1 is already inside H. S1, where we started with S1, S1, Sn, okay. They are intersections of, just intersections of, sorry, yeah, this each SI's are members of curly S, okay. This is the base, this is the basic elements here, all right. But what is epsilon? Epsilon is H intersection S, right. If it is in H also and instead of sub basic, instead of basic, it's a basic open set that will be inside epsilon by definition. Therefore, U is inside S1, it just means that U is contained in the union of all these, this is the second part that we wanted to show. The RHS here, RHS here, one of the members contains U because that element is inside epsilon, okay. So U is one of them. So started with this one. So we have shown that this union is contained inside that one. So that completes the proof of the claim to and therefore Alexander's sub-base is proved. Therefore, we have completed proof of Tyknoz theorem also, okay. One question that immediately opposed to our mind is, what happens to the analysis of Alexander's sub-base theorem for Lindelof property? Remember, we have already seen that Lindelof is not even finite productive. Namely, we have seen that the L, R comma L, where L is the semi-interval topology has the property that the product is not Lindelof whereas RL is essentially Lindelof, okay. But Alexander's sub-base theorem may still be true. Why, why that is not true? What is happening? Is it not true at all? So this is what we want to question. Does the imitation of the proof in the case of compactness work if we simply try to replace the phrase finite sub-cover by countable sub-cover wherever finite, you know, the doesn't admit finite cover is sort of countable, countable, why, where do we go wrong? You can figure out where it is, but here is a concrete example which says that even the Alexander's sub-base theorem, you know, the statement will not be true if you recall, if you replace the, you know, compactness by Lindelof property, okay. So that makes the Alexander's sub-base theorem more important in some way. See, it's just a narrow thing and still he was able to prove here. That is the whole idea, okay. So for this one, I will just quote this xy 3.91.2, okay. But let me give you a little bit of this one. What is it? Look at the family S of closer intervals a comma b, where a is strictly less than b, okay. Then S forms a sub-base for a topology on R, any family of subsets of a, subsets of a given set x will form a topology as a sub-base, okay. But this topology is nothing but discrete topology, okay. This topology is a discrete topology. Why? Because given any x, I can take something x minus 1 to x and other one x to x plus 1, both close intervals. Intersection will be just singleton x. So every singleton x is open mean it's a discrete space. On the other hand, so once it is a discrete space by the way, an uncountable discrete space is not leaned alone, okay. On the other hand, the exercise there asserts that every cover by a sub-family of S, this is S, admits a countable sub-cover. This is for the usual topology of R, okay. With the usual topology of R, you show that take a and b are close, use the topology of R, that is a finite sub-cover. We just show. The finite sub-cover part is just set theory. So it covers R. Even this may be, sorry, this is not finite sub-cover, countable sub-cover. This may be uncountable sub-cover normally, but you can have a countable sub-cover. So this was the exercise. The point is, if you take open intervals a, b, then of course, you know that it is because R is lentil of. You have to use that also. But now we have to show that even if you cover it by closed intervals, it will have a countable sub-cover, okay. So granting that exercise, what we have is the following. Namely, Alexander sub-based theorem is not true for Lindelow's property. There is another remark which I would like to do, not with Alexander sub-based theorem or Cauchy's theorem and so on, but another property for whether something is productive, namely quotient maps. Take two quotient maps and take the product of these maps. So x1 to y1, x2 to y2, then you have a product map from x1 cross x2 to y1 cross y2, okay. More generally, you can ask for families xi to yi or qi is a quotient map. Then you can take the product map here from product of xi to product of yi. It is a quotient map and then you can study the properties, okay, quotient, straight forward quotient may fail. But suppose you take open maps, suppose you take open quotient maps or closed maps or closed quotient maps and so on. So there are a number of such problems. So I will just sum it up. We will not go to deeper study of these things. They are not too difficult or they are easy or too difficult, right. So some of them I will take it in the part 2 of this course. But right now you can observe that openness is finite productive. If you take an open map, two open maps, product will be an open map that is very easy to see, okay. And if you have open quotient, means open and surjective map, that is an open quotient, then it will be productive, no restrictions even you can take arbitrary products, okay. Three and four are much harder. Just if you take arbitrary quotients, just quotient map, even two of them will not be quotient, will not be a quotient map, need not be quotient map, okay. Unless you assume some more hypothesis, okay. So that is an interesting case here, which is needed in many other places also but that will be done in part 2. So we will stop here with productive properties and so on, whatever so far properties which we have studied. So next time we shall start studying some more topological properties. They will be, in general they will be called as what, what is the name, largeness properties. Thank you.


I hate it. Working 40 to 45 hours a week get you nowhere until you answer these questions. Where do you see yourself in the next three to five years? Is this job what your heart wants? Are you happy at work? Just be honest with yourself.


In this video, I'm going to very quickly go over a couple of vocabulary words here. Some of these you might already know, but it's always good to go over these, especially with the new notation that you're going to see. Parallel lines, perpendicular lines, skew lines, and parallel planes. These are the four that I'm going to go over very quickly. Now, if you look over here, if you look over here, we're going to do this with a cube. Okay? So we're going to look at parallel lines and perpendicular lines and all sorts of different stuff with this cube. Now, those here, we have these red arrows. These red arrows right here, those mean that those lines are parallel. Okay? These are some of the parallel notation that we use, so AB and EF are parallel. And then also, these double arrows down here kind of mean the same thing. It just means a different set of parallel lines, EH and FG is a different set of parallel lines. Okay? Now, this down here in the corner by H, that's the same, that's what you would normally think. That's a right angle. It's a 90-degree angle. And then same thing up here with B. This right here is also a 90-degree angle. It doesn't quite look the same here, but again with a three-dimensional object, we're looking at the top of this box, okay, top of this cube. And so it's skewed just a little bit, but what it means is still the same, is that it's just going to be a 90-degree angle just like the one down there. Okay? So we're going to use this cube to go over a couple of these different vocabulary words. So here we go. Parallel lines, it's just like what you normally think. Parallel lines are lines that do not intersect. They do not intersect. Okay? Now again, this is a vocabulary word you've used many times in different math classes. It's nothing, anything different. Okay? Now, what we're also going to look at here is some of the little bit of notation that you might see that might be a little bit different, might be new to you. Okay? So I'm going to choose two lines that are going to be parallel. Now A and B, or excuse me, AB and EF are obviously parallel. Let's choose something else. Something like CG and DH, okay? Those ones there on the bottom, those ones are in fact parallel, so I'm going to use those. So DH, that segment, is parallel, is parallel to CG. Okay? Now notice the symbol that I use here. This is our parallel symbol. Two vertical lines going up and down. That's our parallel symbol. It's kind of a shorthand for parallel. That's what we commonly use. Okay, next, next we have perpendicular lines. Okay? So perpendicular lines are when two lines intersect, draw a quick picture here. When two lines intersect, they form a 90-degree angle. Well, in fact, to be a little bit more precise, four 90-degree angles are formed. One here, there's one over here, there's one there, and there's one there. Okay? So all four of those angles are actually 90 degrees. Okay? So we've got to be a little bit more precise when we go through our definition here. So a perpendicular lines are lines that intersect to form, now this is where that new stuff comes in, four 90-degree angles. Okay? Using a little bit more notation here, this is the angles, angle symbol right there, angles, okay, for plural. All right? So lines that intersect to form four 90-degree angles, so being a little bit more formal with our definitions, giving out a little bit more information, not just that it intersects to form 90 degrees, we form four 90-degree angles, just a little bit of extra there. Okay? So let's look at a couple of lines that could be perpendicular. Okay, this time I will just use the obvious ones here. So H, C, and H, E are in fact perpendicular, so I'm going to use those as my example. So E, H, that segment, is perpendicular to HG, is perpendicular to HG. So notice that this symbol in the middle looks like an upside-down T, okay, it looks like two perpendicular lines, that is in fact our perpendicular symbol that we use, okay, that's our perpendicular symbol. Just like Parallel's got its own symbol up here, perpendicular's also got its own symbol. Okay. Next on the list, skew lines, skew lines. So when we're dealing with three-dimensional objects like this cube over here, there are certain lines that actually do not interact with each other at all. So these lines we call skew lines, okay, so an example of something that doesn't interact with each other, an example like EH, okay, there's one segment here, and then over here, CG, these two lines have nothing to do with one another. This one goes up and down, this one goes back and forth, forward and backwards. They don't intersect, they're never going to cross each other. They're not parallel, they're not the same distance apart. They really don't interact with each other at all. So skew lines are actually very different. Now again, these can only happen in three-dimensional space, so it can only happen with a three-dimensional cube like this, okay. So skew lines are lines, let's use a different color here, let's use a different color, change things up a bit. Okay, these are lines, lines that are not parallel and, and are not, oh wait, I should rewind there, they do not intersect, and do not, intersect is a, is a verb, so I got to use do, not, intersect, do not intersect, okay. An example of this, as we stated earlier, it would be EH and CG, EH and, use my ampersand symbol here, this means and, and CG, that segment there, okay. So that gives you an example. Now again, there are many, many examples of these, even within this cube that we have here, but I'm just giving one example, keep this nice, short and sweet. All right, last but not least, we have parallel lines, or excuse me, not parallel lines, parallel planes, just like parallel lines up here, it's going to be basically the same definition, except for we're talking about planes, and again, we're going to go over the notation just a little bit. So these are planes, planes that do not intersect, two planes that do not intersect. So this would be like lines on a highway, or excuse me, not lines on a highway. We're talking about two planes. This would be like the top of a table, and the floor. Those are two planes that don't intersect, okay, they're both parallel. That's an example, a real-life example of one. All right, anyway, now an example of two planes in this figure. Now what we have to do is we have to define the planes first. Now to do that, we need three points for a plane. One point makes a point, two points make a line, three points make a plane, okay. Notice the logical progression of the numbers of points making the different shapes. So anyway, what we need to do is we need to figure out some points to use for our faces, for our planes.  We've got lots of different faces, lots of different planes to use. Okay, so for this example, I'm just going to use the front face here, because it's staring us right in the face. And we're going to use the back face, way back there, okay. Notice these dotted lines here, I forgot to mention that earlier. These dotted lines here represent the backs that we can't see. Those are the lines that we can't see. Those are behind everything, can't quite see those. Anyway, so this front face, notice the points here, E, F, G, H. Those are the four points that make up that face. But again, I only need three of them to define a plane. So I'm just going to use E, F, G, just stay with the alphabet. So E, F, G is a plane and it is parallel.  Back face is A, B, C, D. As I said before, we only need three points. I'm just going to use A, B, and C. I'm going to use, get rid of that, A, B, and C. So the plane, E, F, G and the plane, A, B, C, they are in fact parallel. This would be our front face and the back face of our cube up there. Okay, that's just a little bit of vocabulary that goes along with the section that we are learning. Again, that vocabulary is parallel lines, perpendicular lines, skew lines, and parallel planes.


Finally tonight, a love story from the racetrack. Josh Peterson tells us more in this week's Northwood's adventure. Love can be found in the most unique places. But who would have thought love could be found at a racetrack? There's something in the dusty dirt track air at Bemidji Speedway where in the Wazoda Mod 4 class, love is in the air. My name's Brian Feta. My name's Nicole Feta. My name's Dean Shaver. My name is Brenda Shaver. Yes, you heard correctly, married couples competing in the same class. From start to finish, these couples are in the race for the long haul. You know, we just would really like to make sure that we finish every race we start. Have success better than what we had last year.  And try not to wreck anything too badly. For these stock car drivers, teamwork is key. Helping each other out both on and off the track. Nicole does our helmet preps. I try to make sure to do all the fluids and tire pressures. I help with the tire pressures too. That's kind of a dual thing there. Personal preference. Yeah. Where winning the race is the main objective for these couples. It's about beating each other through good old-fashioned competition. Once we get on the track, it's every man for himself or a woman. But it is fun. We've always been competitive together. So it's a good time. She's beat me. I will beat him once. That is the goal of the day to see who beats each other. No matter who else is out there, we just really care who beats who. Yeah. While the dirt flies, so does their emotions. Building stronger relationships over the hub of the sport. We've always done it since the day we were married. We worked together, played together. Best friends. It definitely makes your marriage heck of a lot stronger because you're doing it together. You're just communicating. As both cars and hearts race around the track, one thing is certain. That for better or for worse, these couples are already track champions. But this week's Northwoods Adventure in Bemidji, Josh Peterson, Lakeland News. They seem to get along just fine. It's looking good so far.