Coming to you from RGIS Studios, I am Anicia Antoine and this is your Media News Brief. The Cultural Development Foundation is in the process of establishing a National Dance Theatre Company. Director of events at the CDF, Adrienia Frederick, says they are hoping that from the dance company, dancers will be trained to represent and reignite creative arts in St. Lucia. Richard Ambrose, a participant in this year's Charyfesta believes the initiative will give St. Lucia at least a platform to allow the world to experience St. Lucia art. The Government of St. Lucia has announced a number of projects to come on stream. These projects include a nationwide voter rehabilitation project, new properties to be constructed in the country, as well as the Uranora International Airport redevelopment project. Minister for Tourism, Information and Broadcasting, Culture and Creative Industries, Honourable Dominique Fede, highlighted the Village Tourism Initiative soon to come on stream. The initiative will allow for a different type of experience for visitors to the island, one where they can immerse themselves in everything local. The launch is expected once the legislation has been passed and will initially encompass eight villages, including Grosjele, Ancelary and Souffre, and under the Village Tourism Project, each village will have its own theme, such as wellness, arts and crafts, and local traditions. With the help of a low interest financing supported by the Government, property owners in each village will be encouraged to upgrade their homes and properties to make them more suitable as accommodations for visitors. All owners will be licensed and will undergo training to ensure that the accommodations and services are at the same level as they would be in a hotel or resort. St. Lucia saw record figures in 2018 with approximately 1.2 million visitors to the island, accounting for a 10.2% increase overall. Prime Minister the Honourable Alain Shafny said that while the Government is not opposed to assisting Virgin Atlantic, the means chosen by the airline to obtain that support is not in the best interest of St. Lucia. He noted that the amount of money being requested in comparison to the number of seats being provided and the mechanism chosen, which is a direct subsidy, is not a viable option. The airline is requesting the Government pay a subsidy of US$2.5 million per year over the period of three years. Honourable Shafny maintained that the Government cannot adhere to this request, however it remains open to discussions with the airline. The Prime Minister expressed hope that the Government and the airline will be able to reach an amicable decision before June 20th, 2020. This has been your media news brief. Thank you for watching.


Hi, I'm Dr. Daniel Kraft. I'm the chair of the XPRIZE Pandemic Alliance, and I'm really lucky to be joined by Jeff Huber, the founder of OpenCovidScreen and Professor Chris Mason from Wild Cornell Medical Center. We've just sort of wrapped the announcement of the winners for the XPRIZE for rapid COVID screening, for fast, frequent, cheap, and easy COVID screening. And we're going to have a quick discussion about what we've learned, what surprises about, and something about the winners. So let me pitch it over to Jeff, who really got this whole amazing XPRIZE started. Jeff. Thanks, Daniel. Yeah, the genesis of this was recognizing the critical need for innovation around testing that is frequent, fast, and around, cheap, and easy. And the competition has been an amazing success from the initial over 700 teams that signed up to over 200 finalists to the nine winners that we announced. And it was a very rigorous process that included distributed proficiency testing over 200 samples or 200 samples into each of the teams, give us great quantitative analytical data, then rigorous clinical testing that was run by Chris Mason and his team, very expertly, to lead to these fantastic set of winners. Five, they're sharing our $5 million prize purse.  And Chris, we had an interesting process here to test the testing. Maybe run through what the opportunity challenge was and some of the lessons learned. Yeah, happy to. So it was really, as you've said, at pandemic speeds, we had to get things together quickly, get basically aliquots of titrated known pathogens, not only of synthesized amounts of SARS-CoV-2, but other related pathogens that could make false positives on tests or different matrices, like doing it in saliva versus nasopharyngeal swab matrix, like a VTM, or even just in buffers. So we really wanted to build out these plates where we send them blinded to all the different teams and they don't know what's in them and we have to see how well they do. And actually some did extraordinarily well and some did really awful. So those ones didn't go forward. But we really saw that people came through with some really innovative tests from all over the map, really. Everything from testing your olfactory neurons that are popping up in your nose or throat to PCR and antigen tests. Even an MRI machine was in one of them. So really a really broad range of innovative ideas were brought to the table, which is great. And the trick was to be not just sensitive and specific but frequent, cheap, easy. Jeff... Frequent, fast, cheap, and easy. What does that really mean in the context? And what are the implications of that for testing, not just for COVID, but maybe other next pandemics or diseases? Yeah, and I mean, that was part of the inspiration for the competition from the beginning of, it's clear that this COVID-19 pandemic has a significant impact on the world and society. But in a lot of respects, this is almost a practice pandemic. Imagine what could have happened if the lethality would have been five or 10 times as great. And there's potential out there for other zoonotic viruses pandemics to come. COVID-19 is just one in the series. And also we're still battling COVID-19 of other variants that could develop that are even more challenging. So really this innovation was, yes, to have an impact on COVID-19 and given the pace of it, we still have a lot left to go on the runway with COVID-19. It's great that we have vaccines coming along, but with the evolution of variants and getting all seven billion people in the world vaccinated, there's going to be time that testing is really the solution, the thing that provides the safety net to get us out the other side of this. But with others to come, now with this innovation, we're equipped from the beginning to have frequent, fast, cheap and easy testing to do a much better job for the challenges that we'll face in the future. And Chris, putting your scientist and entrepreneur hat on, where might you see the future two, five years from now with the ability to have testing, molecular and otherwise, in our pockets, in our smartphones and connected back to public health? I think it's really a brave new world and a brave, much more quantified and accurate and responsive world ahead of us. So this includes, as we've seen from some of the innovations in this XPRIZE, point of care testing, home testing, things that you can do very quickly, very cheaply, the frequent, fast, cheap and easy. And really, I think it's gonna be that there's gonna be the convenience of it of having your home is not just about easiness, but also it's the safest way to do testing. So instead of having to go somewhere to get a test where you are probably coughing and infectious, you do it right in the comfort and ease of your own home. And so it is gonna bring some of the convenience we've began to view pregnancy testing to the molecular diagnostics realm of infectious disease, which is desperately needed. And I think it also gives people a sense of agency. They can tell when they're sick and they can have a quantified test to say, I'm definitely staying home from work today. I know I'm sick, here's the results, you could maybe share it. And also it helps with the genetic epidemiology. As soon as it's positive, we start to look at sequencing methods. You can capture that virus, use it to design new vaccines to track how a virus is evolving or any pathogen. And I'm hoping that we'll see more and more of that.  So I'm excited to see more and more companies like this deploy tests on the market and it'd be embedded more into the epidemiology as well. And it might go beyond infectious diseases to non-infectious, so that's another element. Maybe to close out, we don't need just nine winning teams, which you can reach via the link we'll be sharing. Maybe what are some of the surprises? What are your favorite elements that came out of this? And what might be the implications for testing and even prizes in the future? Go for it, Chris. Yeah, I guess I'll go first. I think it's really interesting, like the phenotype of anosmia where people lost their sense of smell is unique to this coronavirus. And so I think it's, we don't know if that'll be relevant for necessarily other viruses, but it might be relevant for other diseases because the nose knows, as a lot of people say, it really has an interesting, it itself is a phenotype.  So I think what's interesting is we've now seen things like the use-smell-it test, which has an app and a series of strips and then you can learn a lot about the functioning of the neurons that are right in your nose and also what's happening also chemically in your brain, like how are those messages being received? So I think that was certainly unique to this virus so far, but it certainly can apply to other possible diseases. And I think the thing that really resonated for me is just the diversity of solutions. Part of the design of this competition was in recognition that there is no silver bullet. Everybody wants a silver bullet, there isn't. We need a variety of solutions. We need supply chain diverse solutions. We need things that work for different use cases and the diversity of solutions represented from testing types of PCR based and LAMP based and antigen based and new innovations in the open innovation category and things that are working in the lab and things that are working in the home and instant results and next morning results, all of those together are what it takes to battle and fight this. And I think the collective public health impact is really profound. The goal is to be able to catch occurrences before they become outbreaks and to stop transmission chains. And these are the kind of tools and technologies and innovation that make that happen for COVID-19 and for future pandemics to come. Yeah, spot on COVID certainly been a catalyst. You know, just like spot next to the space age, COVID is sparking a bit of a health age and things like this XPRIZE, which you know, it tries to trigger audacious but achievable challenges and speed them up, you know, has long term implications like the very first I'm sorry XPRIZE help catalyze opening up space. And now we see SpaceX and others and I think hopefully what emerges from this prize and others like it will really catalyze a future of health, medicine, diagnostics that will benefit everybody. So, the closing line of COVID-19. Exactly. So for those of you listening, you can learn more about this particular XPRIZE at XPRIZE.org slash testing. You can learn more about the winning teams. You can help support them going forward from funding to testing sites to regulatory. There's a lot of potential to keep collaborating here or still in the midst of the journey to help address this pandemic and prevent future ones. Thanks for joining us. Thanks.


Good afternoon, ladies and gentlemen and welcome to Bly Street. I'm Michael Fully Love. I'm the executive director of the Institute It's my pleasure Today to be welcoming an old friend of mine the new commissioner of the Australian federal police Andrew Colvin The Lowe Institute's job is to deepen the international policy Discussion in Australia and give Australia a greater voice abroad and in doing that We look at the full arc of Australia's interests and our broader role in the world And of course threats to those interests come not only from state actors But from non-state actors such as transnational criminal and terrorist networks as well To address this threat the AFP maintains an international network across 30 odd countries So when we think about Australians on the front line, it's not only diplomats Soldiers sailors and airmen. We're thinking of it's also police officers Last year with the downing of MH 17 We saw the AFP again at the forefront of Australia's response deployed to the crash site in eastern Ukraine And that deployment built on a history of overseas deployments by the AFP Including building policing capacity in the Pacific and restoring order during humanitarian crises And one of the most famous successful examples of that was the police-led intervention by Australia Ramsey in the Solomon Islands The AFP also has a crucial role as part of Australia's national security apparatus in addressing terrorism and in the short period that Andrew has Held the job of commissioner. He's been very seized of the terrorism issue Not least because of the awful events that took place just a few blocks from here at the lint cafe Andrew Colvin has served in the AFP for 25 years in areas including organized crime money laundering political politically motivated crime and terrorism financing in the early 2000s when I got to know him Andrew was central to the AFP's Response to terrorism including the Bali bombings the Jakarta Marriott bombing and the Australian Embassy bombing He was chief of staff to the former commissioner Mick Kilty and since 2010 He had been a deputy commissioner working in operations and national Security before he was appointed commissioner Andrew has also completed a master's in public administration At the Kennedy School of Government at Harvard ladies and gentlemen It's my great pleasure to invite Andrew Colvin to give his first major speech as commissioner of the Australian Federal Police Thank you. Thank you Michael for those very kind words and good afternoon everybody It's a very full room out there. I think timing is everything in this particular business Can I firstly acknowledge Michael your presence here today and members of the diplomatic corps AFP members some of my Executive colleagues as well as members of the AFP more broadly. Thank you for coming along and supporting This speech today. Can I also just acknowledge quickly the traditional owners of the land on which we gather their elders past and present? And of course the spiritual attachment they have to this place It's interesting Michael hearing what you said in the introduction and we have known each other for a while now And I think it's probably eight years ago when you first pitched to what was two commissioners ago now The idea of a commissioner speaking at the Institute in Bligh Street, and we said it was a good idea then I don't think we anticipated it might take this long and as I say time in is everything But it is a pleasure to be here. It's it's wonderful to get my first opportunity to say a few words in this forum So as I say it is a privilege to be here and share some of my insights and my thoughts As I commence in a role that I'm very honored to be in very privileged to be in but also quite humbled to take on The AFP has a proud and unique place in the Australian law enforcement landscape. There's no doubt about that Not only are we the country's youngest police agency We also enjoy a very unique remit to support national and international police in operations With a mix of diverse legislative frameworks unique capabilities and a very broad international reach as Michael has talked about much of what we do today is topical and much of what we do today is controversial and This puts us at the center of the media cycle. It puts us at the center of public consciousness And I'm sure when I finish speaking today that some of the questions that I'll receive will very much put me on the spot on those key issues We've faced and met many challenges in that short 35 year history And we're now an integral partner and part of the Australian law enforcement landscape We have developed robust intuitive and mutually beneficial Relationships across the globe and to that end I have the advantage that I take over as commissioner of an organization That is respected Successful valued and in very good shape and I can thank my predecessors for that But in my mind as an incoming commissioner as a new commissioner It's not enough just to lead a police force that can simply do a good job today Essential to my goals as commissioner is to question How do I ensure that the AFP would be better placed in five years ten years or even twenty years from now? I think that is a challenge for all modern resource constrained leaders to lead your organization And to leave it better from the position that you founded in So today is a little about where we're headed and not just what we do I'm often asked an obvious question. What are the AFP's priorities? Unfortunately, I find the answer to not be so obvious and I find it a very difficult question to give a precise answer to The truth is from the AFP perspective our priorities are many I have a ministerial direction that was updated just last year Which tells me what the expectation of government is for the AFP? As you would expect and quite rightly it's extensive and it covers a great deal of responsibility But equally when I became commissioner I said that my term will likely be dominated by national security and that's no great surprise It's an obvious result of the environment the very difficult environment that we are now faced with But when I became commissioner just the same I said I do not want the AFP to lurch so strongly towards your national security That we forget some of our core and traditional policing remits if you like transnational and organized crime And that's a lesson that we've learned from the past and the lesson that I've certainly lived through Now if I wanted to complicate the priority question I'd answer I would present a case that places our efforts on organized crime on cyber on international capacity building on transnational crime on our peace operations that we are part of around the world and I'll place them right alongside Cannot terrorism operations and our protection operations and say that they are all national security So in some ways the question is probably moot Everything we do now is a priority It's simply now a matter of how we move between those priorities according to the landscape that we face on any given day And that frankly is the challenge for leaders of any modern policing organization So what matters to me is not the priority question and maybe it's just too difficult for me to give you an accurate Definition for an accurate answer But I have a different set of questions in my mind How do we retain our flexibility our agility our capabilities to respond to whatever challenges next around the corner? What challenges may we face and what skills do we need to meet them? Is the AFP ready to meet them? What is it? But what is the value that we add to the broader equation and can we do business smarter? Now no one can confidently tell me what the next challenge will be and I welcome anyone who wants to try and do that But that's not unique to policing. It's not unique to the public sector In fact, I would say it's a daunting thought for the CEO of any industry to try and predict what may be around the corner As I said my term as commissioner will likely be dominated by national security particularly counterterrorism But what is the next event that will take terrorism off the front pages? Is it a large cyber attack? Is it a significant investment fraud? Is it corruption? Is it a natural disaster or is it a call from one of our close neighbors in a cry for help? Michael's already mentioned this but had you asked me 12 months ago if I could see a situation With the AFP would deploy teams of unarmed men and women to the heart of an active conflict zone in eastern Ukraine With no notice No area familiarity no established links no partnerships To invent to identify and bring home the remains of Australian victims and to investigate the shooting down of a passenger plane I'm pretty sure I probably would have told you that even as a hypothetical exercise That was a bridge too far for me to consider But we did it and as Michael said we did it very well And I think it's important to note for the record that we are still doing it We still have officers at the moment in Ukraine and the Netherlands working on what will be a very lengthy and difficult investigation So importantly the question in my mind at deploying those officers to eastern Ukraine was not about how many police I needed But it was about the right police the right skills the right capabilities and the right support Now that gets me to my key theme for today Too often and very often policing is judged by the numbers Not just the numbers that make up our crime statistics or our key performance indicators But often the number of personnel the number of uniformed officers available. How many sworn police does an agency have? These are our capacities. How many police do I need and they're so often prioritized over our capabilities What skills what tools what capabilities do I need the police? I have to actually have There's no question both are important and I'm constantly reminded that no police commissioner certainly in this country will ever say They've got enough resources to do anything. We're always going to ask for more. I think that's in our breeding But both are important, but it's our capabilities that we need to consider more deeply Now for over 35 years the AFP succeeded on the abilities and the adaptability of our members and the support and the sharing and the success we've had in our partnerships In order for the next 35 years to have that same sense of purpose and direction The AFP needs to understand as well as we possibly can what that future state might look like Now today's demands on national law enforcement have changed just a little since 1979 when the AFP was established and In the AFP in particular has adapted to those changes over those 35 years Largely from within our existing frameworks and settings For most of that 35 years the pace of change was manageable We could adapt and we could keep up But the nature of the threats facing the Australian community today have become more sophisticated They've become far more diverse And the pace of change as we all know is exponential. So the gap is widening Likewise our communities are changing Social cohesion has increasingly linked more to technology than it is the personal interaction High density neighborhoods with like-minded families and individuals where they live work and play within a much more confined space Is becoming the norm and this presents a different set of social values and different challenges for police So if our lifestyles are not traditional, then why should our policing be traditional? To me, it's no longer good enough for us to adapt our understanding of the environment and the capabilities we need Based solely on the lessons that we have learned from our last experience We must now constantly review and refresh our thinking Of what capabilities our police need and what skills are important to meet that emerging environment We must preempt these challenges to the best extent we can and if we don't We run the risk of stagnating in our development and we run the risk of consigning our organizations to be merely responsive Rather than intuitive So in order to deliver what a future capable AFP may look like We need to understand what the future landscape might look like And for that reason i've commissioned deputy commissioner graham ashton the afp's newly appointed deputy commissioner for capability To develop an afp futures paper It's a paper that will be owned by the afp, but it'll be delivered with the support of our key stakeholders and our key partners This futures paper will refresh our current thinking It will be it will build a capability roadmap by understanding our future demands And provide a sense of long-term direction of purpose to the afp in our members And it will be the hook importantly for me by which all of our future planning Both strategic and operational will hang off The futures paper will develop along three key themes that look at protecting australians a core responsibility for For the australian federal police looking at those prevention disruption and community engagement efforts It'll look at protecting australia's interests Understanding where policing fits within that broader policy space Adding government value and introducing concepts such as police led diplomacy, which i'll say more about in a minute And finally and something that is deeply important to me as commissioner is it will look at the health of the afp So it will give us a clarity of mission It will give us safety and sustainability of our business models and our people and in the current environment I can't Impress enough about the safety of our people But it will also say what the afp member needs to look like in the future We plan to challenge the paradigms of traditional policing methods explore opportunities to help shape the environment we operate in And encourage unique combinations of specific skills capabilities and systems That will support a far more effective afp So working with a key stakeholder group The paper should identify as air identify areas and themes for consideration that otherwise we may not have considered The paper will ask direct questions about the role and functions of the afp and the policy settings that currently shape us as an organization It will shape our investment decisions. It should shape our workforce plan our capability plans well into the future It will not be focused specifically on any crime or any crime type But it will look at generic capabilities that we need to do the job In many ways it will be crime neutral if that's how we'd like to see it It will be the first step in understanding what the afp of the future may look like and will better inform the expectations placed upon the afp in responding to that future environment Importantly it will inform what we should measure How we should measure it to ensure that our capabilities are best placed to detect prevent and disrupt criminal activity So let us consider for a moment Two of those potential capabilities that will be looked at as we move forward On the one hand, there are those core key traditional policing capabilities Those that the public would most likely understand and appreciate in a policing organization the skills of our officers And the tools that we use I'll talk about that in a minute on the other hand We should consider further that unique international remit that role that there's so important to the afp How shall we develop and deploy this very scarce commodity into the future? The common view when you align policing with capabilities Is to trend towards and gravitate towards things Rather than capabilities Now don't get me wrong. The afp like all our police cousins love things We love our toys and they're important part of modern Policing and again, don't get me wrong. They will always be An important part of the focus of the afp to have the most modern and technically advanced tools to do the job But in my view will be the softer skills of our officers as we move forward that will actually define who we are as an organization To the modern and future police officer capability extends beyond the computer that they track or follow information with Or the bear cat that you might see on the streets Dominating some in an inhospitable environment We have to become a developed package of elements that ensures the most effective means of supporting our police Ensuring this effective force requires ensuring that our core policing capabilities the knowledge the skills the technology the systems that we all use Are there to support the officers objectives? And to do that the traditional skills or sworn police officers Need in my view to be supplemented by a nuanced and professional set of tailored skills the environment that we face What we need our police forces to look like in the future is highly unlikely to be what they look like now So while identifying specialized capability as a key to ensuring an effective future Does not in any way diminish the vital role of a sworn police officer It acknowledges that the sworn police officer should be viewed as the end user of skills And technologies adapted for a very specific purpose We shouldn't be merely a carbon copy of what we were in the 70s 80s or even five or seven years ago Police agencies must develop and arm their officers with more than a gun a baton to be effective into the future Now the future afp investigative team if I focus on that for a minute May well only contain a few investigators who drive the criminal proof elements that we're taught all about When we join the police force I expect that it will actually contain an expertise mix where the technical capabilities and skills are provided by specialists Cyber technical accountants lawyers could even be psychologists and alike A harmony of police expertise and professional know-how That in many ways will see the police officer become the captain Who marshals the knowledge and expertise of those best place to understand the methods and techniques used by modern criminals But not only that those are the best place to understand the vulnerabilities that have been exploited and what we must do to address them Now sure you say these capabilities currently exist within most modern police forces Particularly international police forces But the move to a truly integrative workforce will lead to a more effective use of the resources that we have available We can no longer afford to view our workforces as a distinction between sworn And unsworn It must be about our capabilities. It must be about our outcomes. That must be about the best way for it to be delivered Police officers and the execution of police powers will always be central to what we do We can't do what we do without that. But from an afp perspective at least it can often only be one part of the equation Now an obvious example that i'm keen to explore and this will be no surprise to most is in technology And our understanding of the cyber environment Digital natives now dominate the emerging workforce Not only did they understand the digital environment They are intuitive enough to understand how it has manipulated where the vulnerabilities are how it has exploited And what we have to do to adapt So they bridge the gap between the current environment and the skills of a traditional police officer It's not a learned skill. I can teach my police about the cyber environment, but i'm only ever teaching them It's not intuitive to a digital native coming through the workforce now. It's natural and they're not intimidated by it So if that's the human capability piece and if the skills of our members be they sworn or unsworn are our bread and butter Then it's the international linkages at the afp that is the afp's competitive advantage That's our value add that i mentioned earlier Of course one area of afp operations It was largely unpredictable 35 years ago that has had to rapidly develop both in its capacity and its capability Is the reach and responsibility of our international network Target destination for transnational crime as australia during part to our wealthy status Our very early use and early adoption of technology And a relatively cashed up society Means that australia's efforts in the region and globally have continued to be pivotal to the success of the afp And policing operations more broadly The afp operates one of the world's largest and most diverse law enforcement international networks regional instability technological advances innovative criminal syndicates And a widening of the terrorist networks have combined with a globalized world to see the afp's international footprint expand To see it become more sophisticated and far more reliant on international relationships than ever before The afp has an international reach that includes almost 100 liaison officers in 29 countries around the world And an international deployment group more regionally based that sees an additional 300 officers based offshore at any one time Now that's a large portion of my workforce For many years now we have built these efforts around a strategy of taking the fight against crime offshore Often to the very places it originates or to the places that it transits through And it's easy to explain these efforts as targeting the criminal enterprise at its origin a successful method for tackling combating crime But a broader view of efforts would see the afp target crime in a much more holistic manner Attacking the criminal environment in other words target hardening those environments that may support these criminal enterprises Now we are responsible for significant capacity development projects And surprising to many we are one of australia's largest deliverers of foreign aid On the surface that's an odd role for a law enforcement agency To actually undertake But in reality it builds perfectly on the strategy to strengthen australia from crime By helping our neighbours develop their own capabilities their own safety and their own security The bilateral and multilateral sharing of information Evidence technology and capabilities Through our international presence assist us to deliver on our promise of leading the disruption and prosecution efforts An interesting fact is of the 2000 odd afp investigations that we have in our national and international arena at the moment Over 60 of those have a direct link or association with international law enforcement or transnational crime The afp is an equal share partner in international law enforcement efforts and we're responsible for for developing And through our network delivering information Evidence technology and capabilities that are world leading It has worked and we will keep doing it But where should we develop this capability and how should it evolve and how can we derive the maximum benefit from the work that we have done Well, this is an answer that the future papers will need to consider and respond to But there are simple things that we can start to consider in anticipation Police lead diplomacy is a concept that utilizes law enforcement links more broadly to build upon and find common bilateral And multilateral ground and diplomatic ground where more traditional exchanges often present barriers What country doesn't want to cooperate on terrorism? What country doesn't want to cooperate on organized crime on child sex tourism on cyber crime and a light The work of the afp and the indonesian national police in the years after the bali bombings of 2002 Was and this is in the eyes of many analysts not in the eyes of the afp Or certainly not only in the eyes of the afp. It was a high point in the broader bilateral relationship between our two countries Now underscoring these efforts that police led diplomacy the building of those broader and deeper relationships Will ultimately work to prevent a criminal's ability to hide in the gaps created between jurisdictions These relationships have to deliver though more than simply good exchanges of intelligence That must evolve beyond strong international cooperation Even beyond the strong operational collaboration that we are currently seeing If we're to build upon this base that we have and evolve our international capabilities along the lines Of police led diplomacy Then we will open up the opportunity to truly make a difference on the impact of crime on austria by harmonizing the laws By working with our partners on better policy By improving legislation By improving rule of law frameworks and a greater understanding of both the criminal cycle And the root causes of crime Now austria's efforts in the solemn islands team or less they more recently pop unigini have been challenging at times They've been frustrating But they've ultimately led to better relationships between our countries better law and governance arrangements in those countries Places where health education and investment initiatives can thrive But from where we see less impact of crime emanating onto austria Of course, this isn't just the remit of the afp It involves a whole of government approach and that is something I can say that the that austria is very good at But we have the relationships and we have the history to make it a possibility The more we do this The more the net will tighten and the less reach transnational crime will have on austria So as commissioner, I kind of hope to achieve this vision unless we continue, of course to perform the tasks expected of us today to an exceptionally high standard Any police force must be judged by its performance in the here and now that's only right and it's proper And we'll continue to do this, but we'll do so with an eye towards the future I do not want us to stagnate nor do I want the afp to be satisfied simply with the status quo To do so we only limit our ability to continue to perform in the future We have a unique opportunity to show leadership nationally and internationally Far from being an agency with responsibility to enforce and implement the laws and policies that somebody else determines I envisage an afp actively engaged in influencing our environment and our future To do so we must first shift the emphasis away simply from capacities And focus equally if not more on capabilities A futures paper for the afp will be the first step in that process So the afp I hope has a bright future. I know it has a bright future We are and we will be even more so in the future one of the most exciting and rewarding employers across any industry With employees who are valued skill Diverse and who are very focused on the national interest and the national priorities It will be challenging It will be ambitious and it will be complex But that I suggest is what we should expect of all of our national agencies of government and its leaders Michael before I finish if you'll indulge me for a further few minutes I'm sure that there is a great desire to hear me speak about the barley nine It's not my it was never my intention to speak about that here today But I recognize that this will be expected given current events. So if I can say a few words I first like to say that I understand this is an extremely difficult time for the family friends and for mr Chan and mr Sookamaran He would not be human if you did not feel for the situation that they are in For many months the afp has been doing what it can to support the whole of government diplomatic efforts And today I would like to again add our voice to the Australian government's plea for mercy It's our hope that the Indonesian government will reconsider its decision to proceed with the executions As the foreign minister has previously said their rehabilitation is a great testament to the success to the success of the indonesian rehabilitation programs in prison I'd also like to say that I understand that the Australian public's right to better understand The afp's role and work back in 2005 during this barley nine investigation is to be expected I'm prepared to say that much of the information that has been circulating in recent weeks Doesn't accurately reflect our role. It doesn't accurately reflect the work that we did in 2005 And unfortunately he ignores the findings of several reviews judicial hearings that have since scrutinized the afp's actions The afp has at all times and will always be Transparent and accountable for our actions and it is no different in this particular matter So at the right time I will discuss this in a lot more detail and I'll take the questions that the public obviously wish to ask But while the government efforts are continuing to help mr. Chan and mr. Sukumaran Now is not the right time for me to go into that in great detail. So with that Michael I'll close. Thank you very much again for the opportunities to speak. I'm sure that there'll be some questions And I'm happy to take whatever questions there may be Well, ladies and gentlemen We've heard a very impressive speech from the commissioner very broad ranging touching on some of the Organizational challenges that he faces as a new commissioner talking about police led diplomacy And addressing the question of the barley nine and the circumstances of mr. Chan Mr. Sukumaran We have the commissioners kindly agreed to take your questions. We have about 20 25 minutes Just before I go to the first question from the audience. I might take the chairman's prerogative Andrew and ask you about an issue you didn't talk about in depth and that is terrorism We as a country Especially where we are just down the road from martin place. We feel The the anxiety that comes from these kinds of attacks on our country But we're also anxious to make sure that Our response doesn't do violence to the values that we cherish as a democracy So can I ask you as a first question as a new commissioner? Looking at this. How do you strike? How do you think we as a country? Are going at striking this balance between security on the one hand and liberty on the other? Yeah, thanks mom and look it's a it's a good question. It's something that's dominated The six months since I've been interim and then and then commissioner obviously Four days after the previous commissioner left. I stood with the Prime minister and then the then director general of asio and we raised the national threat at level too high We did that because We felt that the information that we had available to us needed to be relayed to the public and we needed the public to Be taken into our confidences in terms of what our expectations were What we're very clear at the time was to try and get the message out there that we didn't want people to overreact We wanted people to go about their normal business But we wanted them to be aware so that they could assist police and assist law enforcement And thankfully I can say that that has largely from our perspective worked. We are getting very good cooperation from the community Now events of december last year I think attacked and hit deeply at the heart and the core of who Australians are And one of the things that I found remarkable was the response of sydney and the response of the Australian community more broadly After those events. I mean what we saw Develop in martin play spontaneously with the flowers Was something nobody had predicted in fact We were concerned that there could be a negative reaction And I think that Gets to the core of who Australians are and the way we perceive these threats the way we want to deal with it Very little is going to allow Anybody to shake the fabric of who we are and having just returned from the us I can say that that was commented upon incredibly favorably by a lot of countries the way australia reacted Now are we are we finding the right balance? That's that's difficult The first job of government as the prime minister has said many times is to protect Australians The first job of a police force the first first job of a police commissioner Is to protect Australians and sometimes as you'll see but that will mean that we will act a lot earlier Rather than waiting. I think the threat has changed and morphed in this country so much so that agencies have to change the way they react There's nothing that I want to do either as police commissioner nor as andrew colvin member of the society that wants to pick away the fabrics of who we are Making the decisions that we have to make around putting more Security around parliament house the center of democracy in this country was a difficult decision and not one we took lightly But it's the balance that we have to find between protecting individuals And making sure that those personal freedoms and liberties are also protected The answer to your direct question. How do I think we're going? I think we need to keep watching it I think we need to be very careful, but I think we're doing well So we have about 20 minutes left. I'm going to take questions as I see people put their hands up Can I ask you to wait till a microphone comes? Can I ask you please in the interests of fitting in as many questions as we can to ask a question? Please don't make a statement because I'll rule that out of order. We're looking for questions to the commissioner This is the first gentleman. I saw in the red tie Thank you, michael arens uh transparency international could I Compliment you on taking the lead in the new network for anti-corruption That you've uh, you're heading up in canberra with the cooperation of what I think eight other agencies and uh In particular the new one, which is the linkage with the uh with austrac. Yes Do you see that as a very growing linkage that an effective one for? money laundering anti-money laundering in this in your Under your priorities many many priorities. I appreciate look thank you And uh, so the the team you're talking about is our fraud and anti-corruption center Which we stood up a few years ago and and to be quite honest and frank We stood it up because there were criticisms of the organization about whether we had the balance right in terms of foreign bribery matters corruption investigations We did that in partnership as you say with eight other agencies and austrac is a key partner for us money laundering and flows of Cash illicit cash in this country are quite large. Money laundering has become an important part of what we do I think there's a there's a perception often that Money is often drug money. Um, it's not there's money Generated that comes from a whole range of crimes Some of which are the types of things that transparency international are very focused on around bribery corruption So that is a key part of the work for us and the reason why I was very specific in what I said in my speech is They are that that's important work for the afp. It's work that the Australian community expects of us It would be a shame if I was able to do nothing else But terrorism Now that's a priority and and I can never ever Negate that priority away because I have to protect individuals first and foremost But I will work hard to make sure that we don't lose the focus on those areas like our foreign anti-corruption work In fact, I see it as the future. I see it as a Growing and emerging problem for Australian society Next gentleman I saw here was in the corner Carlos Salmon from Australian Associated Press commissioner It's been widely reported that the afp provided information to the indonesian authorities that led to the arrest of the barley nine Put simply does the afp have blood on its hands? And secondly What do you have any personal misgivings or regrets about the way things were handled? Okay Put simply do we have blood our hands? No Put simply was this part of a conspiracy For greater cooperation that I've seen written about no And I find those comments misinformed and misguided As I said though now's not the time for us to go into a long defense or an explanation of our role There is nothing I would say today Even if I did There's not been put on the record in courts in austria in the federal court when we were challenged about our role To senate estimates processes to media conferences to courts in indonesia So there's no utility in me going through that again now at such a critical difficult time For the diplomatic efforts. So I will answer those questions. I believe we need to answer those questions We'll answer them again But now it's not the day Thank you, sir In the middle. Thank you Jane Margat's abc You talked about lessons learned and um I know that you can't go into detail about the operations of the barley nine, but In hindsight, do you think um that there's anything Could the afp have done something differently that may have avoided these two men being exposed to the death penalty now Look, I'm just not I'm not going to go into it because to answer one question It will take me down a path that is not helpful to the government's efforts What I would encourage people to do is look at the history since 2005 We have put a lot of material on the record including the federal court outcomes when we were challenged about our role We have said publicly Guidelines we acted on and acted on appropriately then have been amended and reviewed and we're reissued by the government in 2009 If people want to understand that please go back and look at it I'm not going to add any more voice to that today because the Clemency efforts need to be the priority are the priority and the afp is supporting the government in those efforts Anthony bubalo at the back Hi, Anthony bubalo from the low institute The foreign fighter problem represents a challenge on very many levels But I wanted to focus on two in particular and get your views on on how the afp is responding to this particular challenge the first is the That in terms of a terrorism problem for really perhaps the first time the significant number of australian nationals involved and secondly the large number of western passport holders Relatively speaking in the front in terms of the foreign fighter numbers and what that and and their ability to move relatively more freely than that perhaps previous generations of International terrorists Yeah, look, thanks Anthony. I think you summed up in in a couple of questions the challenge that we have When I think about where we where we are now and people who know my history will know that I was heavily involved with ct in the you know counterterrorism and the frameworks immediately after september 11 and the bali bombings the threat is different now Largely, then it was an external threat and when it was an internal threat such as the operations You saw in melbourne and sydney ramp and then it's me We were dealing with a confined call that we could we could take action against The threats different now and it's been externalised But it is now becoming a domestic issue and what we what we are seeing is people who are Moving very quickly from thought to intent to action without opportunities for law enforcement and security agencies to pick it up earlier So that's that's the nature of how the problem has changed How are we reacting to that? We're doing what we have to do To work with our partners first and foremost I want to reassure the community again that in this country you have agencies that work very well together But one of the most important parts of it and and we talk to the community all the time Is is the communities and i'm not just talking about individual communities. I'm talking about the community as a whole working with law enforcement to To try and you know help us understand the dynamic that we're dealing with in terms of that broader question around The flow of foreigners. I think you know, there's various figures been quoted about how many Western countries for instance are involved in the is flight flight in Syria northern iraq and you hear all sorts of figures around three thousand That does present challenges It's a it's a difficult policy problem for the government and there's a lot of discussion a lot of thought about As the prime minister has said about citizenship about passports about removing passports We have international obligations not to export Australians to be involved in terrorism Likewise, we have obligations to make sure that Australians are safe back home. So, you know, we are we are Keeping the public very well informed the prime minister is certainly on the front foot about letting the public know What our policy settings are our thinking, but it's a difficult challenge I'll ask you on that how important is the australian muslim community as an ally in the Um, the the fight to prevent radicalization Well, they're they've fun been really important But what I want to be careful too is I don't want it to be all about the australian muslim community. I think they are often Understandably feel targeted Targeted because we want to work with them targeted because they're seen as being In some way involved in in the issues and they're not i'm in australian muslim population Are incredibly supportive of what we do From an afp perspective, we have good outreach and we meet with them a lot We could not achieve the success we have in stopping attacks in this country without them But equally we need the whole community engaged. This can't be about one part of the community And all the focus being on them I feel greatly for members of the Australian muslim community whose children are targeted on buses on the way to school Who are afraid to go to work or are afraid for their wives or partners to do the shopping I mean, that's a really terrible indictment on this country and society And I don't think that's that's not in australia that any of us actually support What did you think as a matter of interest you you talked about the the response of sydney siders and australians to the link cafe Siege, what did you think about the i'll ride with you phenomenon that developed on twitter? Yeah, it was interesting. I mean the it grew very quickly and it grew from a good place My sense is it grew from a good place It it was distorted and misused by some people And people who wanted to interpret it as someone Rising for a fight for instance Interpreted that way but at its core. I saw it coming from a good place And I think it reflected what happened in martin place since so far as the outpouring of support and the outpouring of This isn't an issue for islam. This isn't an issue for the muslim community. This is about criminals doing terrible deeds Thank you the gentleman on the end Thomas london commissioner the recent mckinnon ct review recommended the appointment of a national ct coordinator And profit three options for government to consider The director general of security or a senior public servant either the attorney general's department or pmc Just wondering if I could gauge your view on why wasn't a senior police officer considered appropriate for that position Given the high threat you've mentioned and the very operational nature of today's problem I guess I'll be careful how I answer that because that's a current policy consideration and governments working through How it wants to respond to that and and what the ultimate construct of the ct coordinator would look like but I don't quite agree with your premise that a senior police officer hasn't been considered for that role It's open. It's certainly the director general of asia has been nominated by some Journalists as an obvious person and others have said it's the wrong person That's still something that the government's working through and from my perspective I think a senior police officer should be in that mix because at the end of the day The the front for this issue is dealt with by police around this country Yes commissioner paul male from the australian newspaper, uh, you spoke a little bit about the um The fact that the afp is actively involved in the the whole of government approach Aimed at sparing the lives of these two Australians. Can you just tell us specifically what the afp is doing or has done in terms of of Persuading the indonesians to grant them clemency whether or not the afp has made direct representations Engaged in its own sort of as you call it police diplomacy I asked that knowing that the afp has traditionally been very well regarded up in indonesia And I would assume is is attempting to leverage that goodwill. Yeah, no, we are thanks paul Look, we've been careful not to go into the nuts and bolts of everything that's been done by government There's been a lot a considerable amount done from the afp's perspective I think the attorney may even have mentioned it yesterday in parliament That you know, I've written to my counterpart Along very similar lines asking for the indonesian government to consider clemency and to look at the facts and circumstances of course people forget that Former commissioner mitkelty and my deputy and my now deputy commissioner might feel and actually went and gave evidence On behalf of some of the barling nine in courts in indonesia I think that's kind of been forgotten in the process But the other key part paul of what you say is there is that police-led diplomacy We do have good relationships and we do have influence and we have where possible We are using that influence through those softer diplomacy skills to try and bring about a different outcome Yes, I'll go to aran connelly And then this gentleman Thank you commissioner aran connelly from the lowe institute. I understand you've addressed the ballie nine issue and I want to set that aside But cooperation with the indonesian national police must be very important to you But also very challenging given that this is an organization whose chief and vice chief are embroiled in very serious and very credible corruption allegations and Against which there have been accusations of involvement in both drug trafficking and people smuggling How do you make sure as a general principle in your cooperation with partner agencies overseas that afp assistance? And cooperation doesn't lead to involvement in things that you wouldn't want to be involved in. Thank you I guess the first thing I'd say is the the kpk who was the anti-corruption body In indonesia is also a partner of the afp and we work very closely with them and some of the more recent reports about their their findings and their view of the Kapori the commissioner elect have been widely reported It's a good question because It's it is a challenge. It's difficult. We need to and we have to work with indonesia and the local authorities And we have a great relationship with them But my office is no absolutely that integrity is the number one foundation upon which police forces are built And largely that's not an issue for the way that we go about doing business. It doesn't become a it doesn't become something that inhibits what we do to the extent that any my officers did see Actions or activities that they thought breached what we would consider our core values Particularly around integrity and corruption They know how to deal with it. They know how to report it and they know how to disengage and that's what I expected from gentlemen in the corner Michael Saffi from the Guardian Commissioner given the number of Australians fighting in Iraq and Syria has the afp investigated the possibility of Of liaising with sort of what the people describe as disillusioned foreign fighters with the possibility of bringing them back to Australia And if not, do you think that that's a possibility? They'll have to look into at some point in the future given the sheer number of Australians who are fighting over there Yeah, thank you. It's been reported already that there are Australians who've returned from the conflict zone I don't want to add too much speculation to numbers. I know it's been it's been talked about Between ourselves and the security partners We're very conscious of of engaging those people and working with them where they want to work with us I mean they need to want to work with with police or security agencies in the first place The broader question though is those that are currently fighting if and when they might wish to come back And that's something that's started to be played out in the media. It's a very active policy consideration at the moment I don't want to Reveal what the policy is here and now But it is something that we're very seized of and it's going to involve a range of treatments Now the prime minister has said and are of course standing with him when he said in melbourne when we raised the threat level That the Australian community needs to be protected and we need to ensure that if individuals come back into austria Because they're Australian citizens That we can be assured of the safety of the community and that's our number one priority There's a range of ways that we can deal with that and i'm very interested in engaging with these individuals where they want to talk to us Lady at the back in the blue dress Thank you commissioner You've spoken today very broadly about capabilities and involvement given that what boundaries or constraints if any Does your status as a police force put around your involvement in discussions and decisions of national interest? Okay, if i understand your question right there are there's a lot of there's a lot of restrictions on police agencies Probably more so in this country than any country about how we use technology and what could be defined as being in the national interest So if i take your question to be that that infringement upon civil liberties and freedoms is that what you're Yeah, look there are very there's tight legislation around what we do So everything that i want to do in the capability space and as i said in a speech it's Yeah, there are Tools toys if you like that um a very technically advanced that allow us to do things that people may consider to be invasions of Privacy and intrusion um, you know, we haven't even touched on the current metadata debate But you know that's a that's a hot debate and it should be a discussion in the community It's no question But law enforcement agencies in this country have very tight laws that dictate how we do it my My focus is around those softer capabilities though. I want my police to To have behind them the specialization that they need to get the job done because it is an increasingly complex world You know, I can only talk from my experience as a from 25 years in the in the afp That uh, you know the statute books when I was a young investigator ran to about that thick now they'd fill this wall um You know that what was quite simple legislation then is now incredibly complex legislation balancing the needs to protect Civil liberties and rights with the needs to protect the community and and to enforce the laws So we have tight parameters on that what I want is my capabilities for my police officers to have the skills and expertise to deal with That the way that I expect them to Andrew, can I ask you a question on if In the mid-naughties At the time of the deployment to the Solomon Islands the international deployment group of the afp was really very significant part of your operations And that was an amazing Operation in the sense that it was a police-led intervention the military was involved There was a warship or two standing off offshore there were some soldiers there But it was it was very much police led and I think most analysts would say that the first Couple of years of that operation were very successful. I think then analysts sort of part ways on how much lasting change was delivered to the soloments by the whole ramsey effort But in terms of the uh, the initial Stabilization operation. I think it was very successful Can I ask you what lessons has afp learned from uh, ramsey and can you foresee A time again in the near in in your term perhaps as commissioner when Um, that kind of police-led intervention may be called upon again Yeah, look, I think those police-led interventions are Are critically important and we see all around the world Interventions that are military led and they're successful and they do a lot of good to build immediate peace and stability But my interest is what's the long term? What happens after that? How do you build societies? How do you build rule of law to make sure that a society can function for the longer term? Ramsey is a good example and there's a lot of different views about whether ramsey has been successful or not None of those will actually be able We'll never know who's right who's wrong until the point that australia completely falls out of the solomon islands and says Over to you, um, you know p.m Sort of array or whoever and we see how it goes But there's been indicators along the way and the most recent one was the election held at the end of last year that Again, this is an afp view. It's mostly independent observers was a fair and free election There's always going to be challenges But largely fair and free and from my perspective was done in a peaceful stable secure environment That's a long way from say 2006 elections where there were lots of problems So there indicators of progress and and along that path the military pulled out and its transition from Peace and stabilisation to capacity building by police I think that that's a model that can work not just in bilateral Interventions, but in un interventions as well and we've learned lessons from team or less day where there was a couple of different Starts after the 1999 Intervention and then the 2006 intervention where the un learned and we learned the really Is there is there a lot of value in having 36 different countries trying to teach the Local police force how to police? Maybe we need to have it, you know a handful of countries So I think there's a lot of lessons there. I believe that There's no point having peace and stabilisation unless there's something that comes after and that's often A good time probably for one more question if there's another Hand in the air I'll go to the gentleman at the back as you had a question Paul Thank you for your talk commissioner, but you made no mention of your relationship with the state police forces Which I would think is highly important. Would you like to elaborate? Yeah, absolutely. I don't thank you I don't want anything I've said and there's a couple of my colleagues from the state police in the room There's nothing that I have said today that should be interpreted as As not wanting to work with the state police quite the opposite Today was probably more looking externally to Australia, not internally Our relation with the state and territory police is very good I don't I know you would expect me to say that but I'm not just saying that I've been In the organization for long enough and in the national side of what we do Where from day one I've been working with state and territory colleagues We enjoy a better relationship now than I believe we have at any time in our history Of course, it's always going to be fraught with Issues and concerns, but that's the wonderful nature of federation in this country and the way that we're constructed What's important and there's very little if you look at the success of the afp All of it has been with our partners. So the success of the counterterrorism teams They are joint counterterrorism teams with with state and com of partners If you look at the success of the national anti-gang squads that were stood up over the last 80 months They have been stood up with our partners around the country. You look at the success of our child exploitation task forces They are joint task forces the work we do at the waterfront are joint task forces understanding vulnerabilities So it's it's homogenous. It's it's seamless We work very well with our state and territory colleagues That's the question of the value add that I want to ask What is what value are we constantly bringing into the equation because I only have six and a half thousand employees Andrew Scipione has close on 20,000 Um, you know, uh, tim cart ride in victoria has 16 17,000 So if we're not bringing something to value to the table, then there's probably not a lot of reason for us to be at the table At the moment we bring a lot about you It looked a little bit envious when he said Andrew Scipione has 20,000 employees Lady ladies and gentlemen, um, I think we've we've been very lucky to hear from the commissioner today When I met andrew eight years ago He was very young for the job that he was doing as chief of staff to the commissioner of the afp Now he's very young to be a commissioner of the australian federal police But the reason that andrew has always been picked out in my opinion and has always stood out Among the ranks is for his ability and his breadth and his integrity as a person And I think you've seen some of that integrity and brett's today in the way that he has Answered questions on really every aspect of His work. So andrew, uh, I'm sorry that it took eight years To get a commissioner of the afp here and three commissioners But we're very glad indeed that you joined us. So please join me in thanking commissioner colvin Ladies and ladies and gentlemen, thank you too also for coming Let me remind you the next event of the institute next Tuesday will be on a topic that andrew touched on And that is the question of uh, islamist terrorism and in particular we have a discussion on indonesia and Islamic state which will be led by the very well known international expert sydney jones. That's on next tuesday afternoon I believe so if you're interested, please go on the website and come along. Thank you ladies and gentlemen


So, in the previous lecture, we had we had ended the previous lecture with this exercise. So, let us just do this exercise. So, what do we have? Let us make a picture of this. So, let us say we have R n. So, this in our case let us just take R 2 and we fix this point a comma b and we want to show that the complement of this point. So, we are leaving out this point. So, we want to show that the complement is an open subset right. So, let us just take any point x comma y right and then. So, how will you prove this? So, let x bar m in R n the complement of this right. What this simply means is that x bar is not equal to a bar which implies that there is some i such that x i is not equal to a i right. So, let epsilon be the absolute value of x i minus a i right and this is positive yeah. So, we claim that s epsilon by 2 of x is completely contained inside minus this a 1 up to a right. So, in other words we are taking this distance epsilon this epsilon and we are taking a square let us just to be safe let us just take epsilon by 4 right. So, we are taking this s epsilon to the by 4 around this point x comma y. So, let us prove this. So, suppose if y equal to y 1 up to y m belongs to s epsilon by 4 x. So, then this implies that mod x i minus y i is strictly less than epsilon by 4, but then this implies that yeah. So, this implies that y i it cannot be equal to a i right because x i minus a i is equal to epsilon yeah. So, therefore, this implies that right. So, this proves that this thing inside in the complement given any point in the complement we have found a basic open subset which contains that point and it is completely contained inside the complement. So, therefore, the complement is open. So, this implies that oops I am sorry is a open subset is an open subset. So, this implies that a is an open subset which implies by the definition of closed subset the singleton is a closed subset ok. So, with this let us begin the next lecture. So, in this lecture we will define the closure of a subset yeah. So, let us first write the definition and then we will see what this means by means of some example. So, let a contained in x be a subset. So, it can be any subset does not have to be open or closed yeah. So, the closure of a is defined as follows. So, a closure is equal to those points in x such that for every open set or rather if u is an open set is an open set containing x then u meets a, u intersection a is non-empty ok. Let us try to understand this by means of an example. So, the simplest examples are on the real line. So, let us take the real line with the standard topology and we can take the interval 0 1. Let us take this interval 0 1 and let us see what its closure is. So, suppose we take any x which is not 0 or 1 let us say x is strictly less than 0. So, then we can find a small neighborhood around x which does not meet this interval a. So, a is equal to this open interval 0 1 right. Similarly, if we take any. So, this implies that. So, this implies that if x is strictly less than 0 then x does not belong to a closure yeah. Because the closure requires that every neighborhood every open subset which contains x should meet our set a right. And similarly, if I take any x which is strictly greater than 1 then also we can find some epsilon small enough such that the epsilon neighborhood does not meet a. So, once again this implies that if x is strictly greater than 1 then x does not belong to a closure. And let us make a remark maybe I should have made the remark here itself. It is completely obvious it is clear that a closure a is contained in a closure ok. So, let me just highlight that. So, therefore, a closure. So, in a now in this example. So, in this example a is contained in a closure and a closure has to be contained in this closed interval 0 1 right. That is what we have seen because if we take anything which is strictly less than 0 then it is not in the closure. If we take anything which is strictly greater than 1 then also it is not in the closure the only possible points which are which could be in the closure are 0 and 1. So, let us see if they satisfy the definition of a closure of being in the closure. So, if we take 0 then no matter which neighborhood of 0 we take if we take any open subset. So, if u is an open subset which contains 0 right. Then by the definition of the topology then there exists an epsilon positive such that d epsilon 0 the ball of neighborhood radius 0 epsilon around 0 is contained in u right. But clearly epsilon by 2 belongs to this ball and epsilon by 2 is contained in 0 1 right. So, this implies that u intersection a is non empty. So, therefore, 0 is contained in a closure and similarly one can check easily that 1 is contained in a closure. So, therefore, a closure in this case is precisely the closed interval 0 1 we can give a slightly more complicated example. So, let us give an example in R 2. So, we can take this region right. So, this region is a is equal to those x comma y in R 2 such that x square plus y square is strictly less than 1 and I would not write the details, but you can check that if we take any point x comma y a comma b let us say. So, that a square plus b square is strictly greater than 1 then we can find a small neighborhood which does not meet a there exists epsilon positive such that s epsilon a comma b intersection a is empty right. And similarly we can check that if a square plus b square is equal to 1 then for every epsilon s epsilon a comma b intersection a is non empty right. So, this will show that a closure is exactly the set x comma y in R 2 such that x square plus y square is less than or equal to 1. Note that a is obviously containing a closure. So, a closure is just adding the boundary this boundary circle ok. So, as we proceed we will get more familiarized with this notion. So, for now let us prove a lemma. So, although we will write a proof it is good to keep a picture in mind while we prove these statements. So, let a contain an x like for instance a picture in R 2 will be good enough be a subset a closure is closed in a is closed in x. So, that is why the word so, this justifies ok. So, let us prove this. So, it suffices to show x minus a closure is open right the definition of closed subset was the complement should be open. So, that is what we are going to show that the complement is open. So, let us pick. So, let x be an element in x minus a closure right. So, that is what this means that x does not belong to a closure, but then by the definition. So, then by definition there exists an open set u containing x u contains x or which contains x which contains x and u intersection a is empty right, but it follows that if y is any point in u right. So, then y has an open subset which is u itself which does not meet a yeah has an open subset namely u which does not meet a. So, thus y is not contained in a closure right. So, therefore, we have proved that. So, this implies that u is completely contained inside x minus a closure. So, thus for every x x in x minus a closure we have found an open set u sub x which is which contains x u sub x is contain x minus a closure right. So, and so therefore. So, this implies that this x minus a closure we can write it as union x minus a closure is u sub x right and each of these is open and an arbitrary union of open sets is open. So, this implies that x minus a closure is open ok. So, using. So, yeah. So, roughly this says that if we take any point here we can find this small neighborhood which does not meet the closure and similarly if we take any point here and the complement which is this open region in red is open ok. So, let us prove this proposition the next proposition. Let us use this lemma to prove the next proposition a set a is closed if and only if a is equal to a bar or a closure right. So, let us prove this. So, recall let us just recall the definition of this a bar. So, x is in a bar if it has the property that given any open subset u which contains x it should meet a. So, let us prove this. So, let us assume first assume that a is closed. So, we need to show that a is equal to a closure. So, since a is contained in a closure we already know this is obvious we have to observe this in the remark which followed the definition it is enough to show that a closure is contained in u. So, taking complements. So, we should prove this to show this it suffices to show that x minus a closure contains x minus a and which is what we are going to prove. So, let us take let x belong to x minus a. So, as a is closed it follows from the definition that x minus this implies x minus a is open. So, let us denote this open subset by u then. So, thus there is an open subset which contains x and such that u intersection a is empty right u is defined to be the complement of a. So, therefore, u intersection a is empty right. So, thus x does not belong to a closure a bar if you like yeah by the definition of this a closure. So, this implies that x belongs to x minus a. So, thus we have proved that we have proved this and therefore, we have proved this right. So, this implies that a is equal to a bar that proves one direction of the proposition. So, for the other. So, next let us assume a is equal to a bar. So, then by the previous lemma. So, what is the previous lemma say? The previous lemma said that a bar is closed in x right a bar is closed in x and since a is equal to a bar. So, thus a is closed in x which is exactly what we wanted to prove right we wanted to show that a is closed. So, this completes the proof of the proposition. So, as a corollary as an easy corollary. So, let b be a subset of x. So, then b closure closure yeah. So, closure of the closure is equal to b closure. So, if we take closure then taking closure again makes no difference. So, let us see how to prove this. So, proof. So, from the lemma. So, the lemma says that this lemma over here it says that no matter which subset we take the closure is always closed in x right. So, applying this lemma from the lemma we get that b closure is a closed subset and this proposition yeah the above proposition says that b closure is closed implies b closure is equal to b closure closure closure implies that b closure is equal to b closure closure. So, applying the previous proposition a is equal to equal. So, we will end this lecture by with two exercises with two easy exercises. Let a contained in b b subsets then a closure is contained inside b closure that is the first exercise and the second exercises. So, let z contained in x be a closed subsets and let a be a subset of x which is contained in z yeah. So, then a closure is contained in that ok. So, let me make a remark. So, exercise 2 says that a closure is closed in x. So, exercise 2 and the above lemma imply that a closure is the smallest closed subset containing. So, we will end here.


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