In March 2020, Europe's universities went 100% digital. Overnight, university education was fundamentally and irreversibly disrupted. So how do we reimagine higher education beyond the pandemic? How can we develop a vision for higher education that strengthens the connection between research and education, developing new and appropriate tools to expand the reach, formats and impact of our research-led pedagogic offering? In the future, change will come in six areas. One, the future is not and must not be all digital. The disruption has highlighted the social importance of learning, the impact of isolation on wellbeing, the depth of the digital divide and the need for flexibility. We need to think creatively on the balance of blended models. Technology will be embedded in the university of the future, from infrastructure to digitally enhanced lecture theatres to wearable technology. We need to develop new ways, however, to enhance what remains the mainstay of most universities the social learning experience. Two, research-led universities should lend their distinctive strengths to lifelong learning. Digitalization is among the most powerful forces of societal and economic transformation, requiring more and more people to learn new skills throughout their adult lives. Universities need to strengthen their engagement with lifelong learning and the emphasis on flexible designs. Developing micro-credentials that are recognized across the centre and beyond could be an important start. But universities must not compromise on their core mission of educating for active citizenship and long-term societal growth and wellbeing. Universities must be enabled to respond to this challenge according to their own strengths, notwithstanding the need for harmonization. Three, pedagogic innovation must be accelerated to educate for continuous change and disruption. The sense of urgency to be able to cope with and lead a changing world demands education that encourages students to be curious, active learners. Students need to be open-minded, able to address complex problems. Digital tools, new and established pedagogies are all needed to enable students to explore and apply their learning. However, diversity of needs must be a core part of the learning designs of the future. Universities have the experience and expertise to pave the way if policy, funders and regulators enable them to do so. Four, we must move beyond red tape to develop enabling and flexible regulatory frameworks. Current quality assurance models cause barriers to international collaboration, despite the best intentions of the Bologna process. To be sure, collaboration cannot do without standardization processes which need to provide reassurance. A key challenge for the future, however, lies in how this need for standardization can avoid bureaucratic procedures that are time-consuming and deeply risk-averse. We need quality assurance models that build on trust and enable. Five, we must articulate the added value of international collaboration. We're rediscovering the value of collaboration at the regional, national, European and global levels. Yet rarely is the value proposition explicitly articulated. Collaboration between different types of institutions, subjects and at geographical levels can provide new and unique opportunities for students. But these need to be spelled out to determine what type of collaboration adds value in which circumstance. Six, we must invest in the sustainability of pedagogic innovation. Educational vision and change cannot come without investment. The pandemic put enormous pressure on staff and students and has shown that redesign is both possible and resource-intensive. We need a viable and pragmatic model for better recognizing the time and effort spent on education innovation. Universities have always changed with society. What's new is the speed of the digital transformation accelerated through the pandemic and the complexity of global problems. They require universities to strengthen research-led education and their students' global skills and mindsets. What does change look like for your institution? How can universities build on their strengths to embrace transformation in new ways? Join us and be part of the conversation.


So we're going to talk about the digital euro and for this I'm happy to welcome Ulrich Binsal, the director general at the ECB, who will speak about the report that the ECB published on this. So we will see Ulrich is with us. Yes, thank you very much Elin and thanks to the panelists also from my side and to Fiona. It was great to listen to the discussion and I try now to share my screen, right? I don't think I'm enabled, I try but I'm trying, I'm trying one second. You are being made the presenter now, so you should be able to now. Okay, yes. So you can see the presentation? Yes. I can maybe, is it better now? Is it big? No. Okay, I have to share the other screen one second. Sorry, give me one. Let me maybe continue like this. I have three screens. Okay, maybe let me start if it's big enough to see. Okay, so digital euro and transfer of payments, the panel already started really discussing this. So we have new kids in town. We heard about of course big techs entering the market. We have EPI as a very important initiative and then we may have a digital euro maybe sometimes in the future. So as you know, we published a report just I think on 2nd of October. The ECB published its digital euro report which explains its thinking really and of course my presentation will be based on this report and I mean let me not go now through the presentation overview. Let me start directly again with what do we mean by a digital euro. It's a central bank liability or call it central bank money available in digital form for use in retail payments which is of course for citizens but also for non-financial companies. Again one more time it would complement and not substitute cash. I think the ECB is saying that at every occasion so Monique's plea I think is fully taken up. Of course some people may see I think gets close to conspiracy theories saying that the digital euro is at the end a mean to discontinue bank loss but that's really not the case. I think the communication is very clear on that. Secondly there is the idea to maximize synergies with the industry not to reinvent the wheel not to reinvent many things that the industry has successfully deployed over the last decades and which the central bank should for efficiency reasons not replicate and then yeah the digital euro has not been necessary so far. There are scenarios under which it will become useful and maybe necessary and that is explained in the report what those main scenarios in the view of the ECB would be where this assessment would change and again as previous speakers here have been saying and Fabio Panetta has been saying and our president has been saying the ECB the euro system needs to be prepared because the scenarios under which then an issuance of digital euro makes sense can materialize maybe even faster than we now believe. So yeah what are the main benefits of a digital euro in short the long version is of course in the report. So first the digitalization of the European economy I think that has also been mentioned several times in the panel. The EU digital single market agenda is very consistent with that idea now that we have a digital euro serving the economy so maybe this point goes even beyond citizens and consumers it would be a tool also for the economy and their terms like the programmable money or programmable CBDC are mentioned in industry circles as also you know being supported potentially by a digital euro. Then there's clearly the scenario of a declining use of cash as a means of payment again you know regardless of the central bank's commitment to issue digital sorry non-digital cash so bank without time limitation still you know usability of cash can go down and Sweden has been one example for that so if we extrapolate this development to the euro area then let's say the usability of central bank money would decline even if we would issue a cash even if the stock of cash would continue to be significant it wouldn't you know guarantee that that it can be used moreover you know just from the growing share of e-commerce and things where already now you cannot use cash obviously you may want to preserve let's say the ability of citizens to rely on central bank money as a further option in addition to commercial bank money and last but not least and that has been also mentioned many times in the panel and in the speeches tackling sovereignty concerns relating I mean he had mentioned foreign CBDC which is maybe also a futuristic let's say concern but more concretely as the panel members also insisted on that there are already now let's say there's a dominant market share and market concentration of non-european retail payment providers in an area let's say of high strategic importance of high importance for citizens for the economy so their digital euro would be another let's say pillar of sovereignty in addition of course to a european private sector solutions that we talked about in the in the panel so we the euro system needs to be prepared to be ready to deploy those benefits in particular under certain scenarios and the euro system vcb is not unique of course in this so it's a global let's say phenomenon of central banks to study cvdc so a bis report found that 80 percent of central banks globally are engaging in cvdc work and here just to mention you know big the big players the people's bank of china has been one of the early movers here and started to study cvdc in 2014 it started a life experimentation and a local and focused deployment of cvdc in the 2020 and here what ravier santa maria has been saying rom has not been built in one day seems to apply also for cvdc looking at the time the people's bank of china has also taken to work on this and the federal reserve system has also let's say joined joined the club recently we have joined the club and another let's say interesting maybe publication is one by the bis of a group of seven central banks i think that was published also on the 9th of october so it gives you some let's say common views of seven central banks including the fed the bank of japan and the ecb on cvdc including on international questions so here that is just to put again the you know the nature of cvdc next to you know the one alternative and one thing that is not an alternative so cvdc is a liability of the central bank it is you know that is a differentiation to a private means of payments which are a liability of a private entity and of course you had already now central bank money as you know cash in the form of bank notes was accessible to the general public and there was also a digital form of central bank money which were deposits with the central bank with the euro system but that access to this type of deposits was limited essentially to bank snow so in a certain way cvdc combines the two in the sense we have a general public access but we have a digital the digital form so private entity liabilities commercial bank money is well known is available in digital form is deployed in in many variants of course you have e-money you have you know payments with cards you have then maybe as an intermediate I think moving further away from commercial bank money is a stable coin projects that entail a liability of an identifiable entity and that is no longer the case if you move them to the right so bitcoin pure crypto assets is not a liability of anyone it is just an asset it is not linked in terms of value to any monetary unit that therefore fluctuates and is not let's say has only to a very limited extent the properties of money and is something therefore completely different so what has to be done towards you know being able to issue central bank digital currency a digital euro so maybe in this in this chart I would suggest to start with the lower middle panel so we have no sorry with the end user perspective and the legal issues no that's really the starting point of course everything has to take place within a legal framework and and with or without let's say further legislation to be confirmed and then we have the end user perspective which of course has to drive everything a cbdc should have you know maximize the usefulness for the user there has to be a value proposition for users so the end user perspective has to be really studied and understood then we can move to design decisions and the design decisions then you know first of all the functional design decisions and then on that basis we can move up to then the technical design no and the technical design if you want has three elements a back end infrastructure and a front end infrastructure and then you know various let's say issues relating to the distribution framework really the front end that will be the access point of consumers or firms there will be the question how digital euro and commercial bank money will be exchanged against each other so all those things then need to be studied in detail and yeah they are quite multi multi-dimensional and and non-trivial so very careful analysis is needed here and the question again how will this new kid in town really or how would it interact with the private initiatives and the cbdc really feels the digital euro is a compliment and not a competitor primarily to private initiatives so so why is that so first of all you know the payment the digital payment market is a is a gigantic market and it is a growing market and if we assume you know the scenario that digital payments will take another increasing share in total payments then you know there's ample space for coexisting in the same sense as there there's coexistence now at the point of sale between cash payment and digital payment yeah so let's say the digital euro would come into play also when anyway there would be more migration from cash payment to digital payments so in so far crowding out of private players is not a necessity at all in this growing market in which the usage of banknotes is shrinking and then I think there's no doubt the private sector will remain more innovative and and therefore should you know always be ahead maybe in terms of its solutions to the more let's say conservative or a central bank solutions and maybe the focus of the central bank solution on on really core services not you know private providing the whole set of value added services that the private sector is looking for I think the central banks and dcb in its report has been clear that there's no ambition to take over the front end so there's the idea that you know synergies with the private sectors should be maximized in terms of the front end solution and then also there's you know this question of will there be a shift of a balance sheet from commercial banks to central banks when let's say massive amounts of deposits which are now with commercial banks would become you know would be shifted by holders of those deposits to to become a digital euro and there again the central bank has no interest no ambition to see that happening because you know banks play an important role in the transmission of monetary policy they are the ones who are able to do or to or they are the experts in credit provisions the central bank has no ambition to substitute that so the balance sheet of banks should certainly be preserved and the functional design of cbdc has to include this you know possible tools to avoid that cbdc becomes such a success that it would crowd out in any way central bank balance sheets and yeah the idea is that cbdc if relying on on the private sector in its distribution in its front end could preferably be offered through supervised service providers obviously you know it's a it's a tricky and responsible activity and yeah it would be natural you know that this is left to the supervised parts of the industry one maybe a crucial um design question is the one account based versus bearer based or token based um digital euro so the idea is very simple account based mean there's somewhere you know a central ledger which captures uh who holds what basically and any transaction is recorded in the central ledger where the bearer design would be you know much closer to the idea of a physical bank note that is exchanged directly between two holders and there's no central ledger recording of this of course the the account based is what you know is really tested and used in the payments industry at all layers from the central bank to the end user in various you know variants and layers um and for electronic payments it's more obvious it relies on online payments to link everything of course to the central ledger where the bearer is more innovative more tricky and also in terms of security of course um creates additional challenges but here really the um the analysis is open-minded to both so in the next phases of our investigation we will continue to look at at both options whereby um in my intuition at least the account based approach will in any case play an important role while the token based the bearer based can play a role in certain situations where you really want to have offline ability it's time to vote so um maybe Elin can explain what the question is on the table yeah so on your right hand side you will now be seeing the poll question the question is which approach to settle the digital euro payments do you find most appealing a account based b bearer c a combination of both so if you click your votes you have 30 seconds to do so yeah please please vote maybe vote even more than during the panel yeah so we have 20 seconds left and then we will show the results on the side yeah when the time is up the results appear so let's see it's like uh the winner of the oscar everyone is excited now we need to see do you see uh the results Elin like oh they're there here they are okay so yeah quite a heavy participation very good so there is a majority for the combination of the two sorry that I spoiled with my view also maybe the question but um I think that's that's pretty plausible then there's a account based um is ranked second and bearer based is the third ranked amongst participants I think um let's say tech um or let's say people who who view cbdc very much also in the context of a dlt of let's say fundamental innovation of of ledger systems they tend to sympathize of course also with a design which goes into the direction of a pure token um and also people who like um you know who who emphasize the advantages of decentralized purely decentralized framework in terms of privacy tend to sympathize I think with bearer elements okay thank you I can then move on in my presentation so here is just a list of uh maybe more concrete um features how um maybe I will close it so how would the digital euro look like concretely that's a good question though so for so far I mean where it has to be worked out but the intuition is that it will look like any other modern payment solution for point of sale and online payments so if it can be embedded into existing private solutions that would not be bad of course and then there's nothing speaking against you know having a very similar appearance of what consumers know from the electronic payments they are currently doing of course it should be available throughout the entire euro area it's so I shouldn't serve the needs of all segments of the population in a non-discriminatory way and curtail financial exclusion for the unbanked and for vulnerable groups so the usability should have really in mind the entire population and efforts should be undertaken to achieve this inclusion of course inclusiveness is not free it's costly to achieve it but that that investment should be done it should be the digital euro should be mindful of privacy and that of course is not limited you know to a bearer based CBDC but also an account-based one can you know ensure privacy to a very large extent and that of course is different from the business models also of some private providers in payment services who have realized the value of payment information in understanding individual consumers demands so that is something that would be excluded for the digital euro by definition it would be risk-free central bank money and it should be free of charge and for basic uses by payers again similar to existing forms of central bank money usable for citizens namely bank notes yes already again time to vote so eline what is the next question yes so the next question is on the screen so this is a multiple question also multiple answers are possible so you can click more than once the question is what is the most important feature that a digital euro should offer a pan European reach b privacy c use with smartphone at the payment terminals d use offline e additional no additional cost f use with the dedicated physical device so here you can you can press more than one you can press as many as as you want yes you can okay okay so if i would be if i would be you i would press everything but that is boring so don't press everything don't press everything so as soon as the time is up we will show the results 20 seconds left so everybody is clicking yeah yeah so you can't think too much anymore there is 10 seconds left okay here we go let's see what the system here we are okay it's uh yeah we have a winner yeah it's uh yeah almost ranked but the winner is a pan european reach okay the the audience is european oriented that's very good and we have next is the use with smartphones yeah we are all smartphone addicts and we want to pay with smartphone and obviously biometrics is i would say the you know the big difference that smartphones also offer now in the payment convenience so that's not surprising then the privacy is ranked very high use offline as well yeah so sorry no be precise next one is no additional costs then comes use offline i think we are almost you know all of us are always online those days but we still dream of being offline sometimes so that's why it would be nice to have a cbdc which works offline for 133 respondents and then yeah the use with the dedicated physical device is not so popular yeah a dedicated physical device maybe we don't want to to have one no we want to maybe use our smartphones and not be bothered with with something in a in addition that's um so that's a negative vote maybe against a dedicated device which you know i think comes into play potentially for offline payments though but okay maybe offline is possible without a dedicated device okay thank you then i'm coming i think soon to the end so further assessment is is needed again it's a it's a project as as big as building Rome almost so hopefully not so now we have you know the milestone the report was published and they are now let's say two in particular two work streams awaiting us which is a public consultation and and further experimentation and i think also that's maybe needless to say conceptual work and and some you know conceptual work issues are listed here yeah um so implications of cbdc for key areas of interest of course for the central bank and and for others uh is uh you know the impact on monetary policy the impact on on the economy therefore and on financial stability and the financial system the banking system so that is uh is a topic often mentioned and you know the the objective is clear of course cbdc must not you know put in question financial stability it should not of course crowd out the private sector in any way so they're clear you know already objectives formulated in the report but to be worked out exactly how those objectives are fulfilled through a functional design of cbdc then yeah the question of privacy of remuneration needs to be worked out and the already mentioned uh technological aspects like for example decentralized um working versus the central ledger the offline usage but many many other questions so then the public consultation yeah um the public consultation has been launched i think was it on 12 uh 12th of october and it will run for three three months so i think it will end um on the is it 11th of january or 12th of january midnight yeah it will end and it includes it's on our website everything is explained there at the end the answers have to be given online you have to register and it it should really address both citizens and and professionals so everybody can answer all questions of course but some of the questions are probably more for more taking the citizens perspective and others are taking the the service provider the financial industry perspective so um yeah i think we we are very keen of course to get input to really understand the requirements i mean from citizens what are the user requirements from from citizens what is the design and from professionals of course ideas on um on technical solutions ideas on the integration into existing payment solutions yeah we are we are very keen to get any idea here uh what what will lead to to really efficient and user friendly um solutions and uh yeah then we have experimentation so if uh you know this is experimentation with maybe a six months horizon so it is not you know experimentation probably with a large-scale deployment of cbdc to let's say citizens for certain uses as it is currently taking place in in china we are not yet there but there's a lot to experiment to check you know for scalability of solutions for the possibility of offline you know there there are various things we also you know can check what our current uh infrastructures that we have can possibly contribute as you know we have tips as a instant payment system so instant i mean cbdc is by definition you know needs to be instant so the question is uh is uh is tips you know can that be an inspiration for a backend solution for um cbdc so you know various experiments of a let's say smaller or medium scale can be done to really um move on you know in terms of understanding the technical options while doing in parallel more conceptual work and collecting the citizens and the professionals input through the public consultation so that's um that's what is currently then um on the um let's say agenda for the first half of next year and then in mid 2021 the euro system could you know consider whether to launch a digital euro project so to really move you know one step further and again it would be launched to be prepared so launching a digital euro project and let's say investing the resources to really become able to issue a digital euro should not be you know mixed with a decision to really issue the digital euro yeah so that decision is not to be taken in mid-21 but the decision to launch a project which you know would allow us or would allow the decision makers to um one day decide to issue the digital euro okay thank you thank you very much and i understand elin we may have questions i don't see them i see them okay excellent so there are a number of questions i um of course i see the time but maybe uleg you have a few minutes just to answer a few of them um so there is one here um that says is instant payment in central bank money in essence not already a transaction in central bank liability the question is of course are they speaking about instant payments as instant payments and tips or instant payments as in rggs yeah so um let's let's take instant payment in tips yeah i mean here from the perspective of the um the uh the citizen the consumer that is still you know a payment in commercial bank money the the settlement is um then instantaneous in central bank money but in terms of the claim of the bank um to um to let's say the central bank money pool in uh in target two so it is uh set it in central bank money but not at the the holder the the payer and the payee don't pay in central bank money they pay really with um commercial bank money so if one would want to use tips as uh the basis if you want the central ledger of uh digital euro then um basically the citizens the users would have to get their own tips accounted in one way or the other so that would be something different thank you uleg and there's another one here what is the role of commercial banks for a digital euro so is there a role for commercial banks if yeah yeah the role i mean that uh that um that what that's you know implied by the idea to rely on on the private sector in in particular providing the front end and of course i mean who who has currently um most european citizens as clients those are the banks so um you know customer identification um and for you know AMA CTF purposes um that's all you know what banks are doing and um i think on boarding citizens to CBDC would would be much more feasible and efficient if it would rely basically on the bank um the bank relations with uh with their customers of course it does not need to be the only way to make CBDC accessible but in terms of ease and scale that seems to be an obvious choice and of course if the european banks um you know launch epi deploy epi of course you have then the economies of scale and efficiency from again a single payment solution that could you know have then even more synergies with CBDC that would be a kind of uh yeah maybe most efficient scenario but again CBDC should probably not be exclusively linked you know to one distribution channel but of course the the bigger the more efficient um those distribution uh channels are that can you know team up with CBDC the better certainly excellent thank you very much so maybe just one more um one more question we have a lot here in here but let's just take a last one could the discussion on CBDC digital euro create uncertainty that might discourage the industry from working on pan european payment solution yeah that that uh the answer should be no and we should do everything uh that everybody agrees that the answer is no because as as mentioned you know we are talking about um a further growing market a gigantic market no retail payments electronic retail payments is growing and is also growing because payment in central bank money in the form of bank notes gets gradually substituted by a digital payments no therefore you know the central bank entering this market does not imply the necessity for crowding out private players and um i mean that's that's a general remark and second there's the idea of teaming up on the front end solution and third there's no ambition to substitute balance sheet so all those things are very clear in the mind of the central bank and we must of course convince the private industry that therefore their investments into private retail payment solutions are fully deserved yeah okay and then this would be it no no further questions no i think otherwise we can be here until tonight so thank you very much it was very interesting to hear all the insights on the possible digital euro and just to remind everyone on the date as we mentioned it is the 12th of january 2021 um so before we close i want to hand over uh to jona who has agreed to do the closing mark of today so maybe Fiona you are there yeah thank you okay thank you very much Aline and uh thank you also to Ulrich for that interesting presentation on the digital euro so uh indeed as the conference comes to a close i would just like to briefly recap on some of the highlights that we heard um we started of course uh by looking at the challenges and also the opportunities of the digitalization of the european retail payments market and uh we have done so under the new normal that the COVID-19 pandemic has introduced uh a situation where jonas at which all in our race and may change one and digital in a matter of just a few months now i don't want to suggest that we expect the european retail payments market to change towards being 100% digital so quickly but still it gives a sense of perspective in in all of this um mr fabio penetta ecb executive board members started this morning by outlining the comprehensive response of the ecb to the changing payments landscape a response which consisted of four elements first our cash 2030 strategy which is to ensure that bank bank notes remain widely available and also accepted as a competitive reliable payment instrument and as a store of value our retail payment strategy which is aimed at fostering pan european payment solutions an innovative european ecosystem for payments as well as full deployment of instant payments and then he also covered the ecb's approach to act both as a catalyst and to foster and support change in the market as well as our role as an operator of pan european solutions such as a target instant payment settlement service tips our new oversight framework was the third aspect that he covered yeah this is a framework covering electronic payment instruments schemes and arrangements the new framework will be called um pisa and this will be released for public consultation in the coming weeks and fourth possible launch of a digital euro that would both shape and promote the digitalization of payments in turn supporting the digitalization and modernization of the european economy and then following on from uh fabio uh speech we had commissioner morade mcginnis who spoke about the commission's retail payment strategy for the eu and uh the different objectives that this wants to achieve first being the broad and diverse range of payments secondly an objective for instant payments to help reduce our dependence on the well-established players such as international card schemes and also more increasingly also on technology companies and third and finally the fact that the commission is committed to enhancing cross-border payments with non-eu jurisdictions in order to make them faster cheaper and indeed also more uh convenient and uh i would like to thank in particular um miss mcginnis for agreeing to speak at our conference because as was mentioned earlier she started in this role only last week on the 12th of october as the new commissioner for financial services financial stability and capital markets union um the two keynote speeches they show the alignment and cohesion at european level and highlight the importance of the efficiency of the european payments market and the sovereignty of the euro uh we then had the panel discussion where stakeholders from the industry and the central banking side discussed concrete actions taken to achieve the basis for pan european payments solutions and what is still needed for these solutions to be deployed and rolled out across europe and we also heard the consumer perspective and what is important for the consumer in all of this during the panel it became clear that there's a lot to be done and that this needs to be done quickly and we have no time for complacency in any respect but overall the elements are there for a pan european or more pan european payment solutions and there's a very definite momentum behind this to go forward and also of course to bring consumers and their needs with us throughout afterwards we had the presentation on a digital euro and how it could impact the retail payments market but from Ulrich Winsall the digital euro would be a new payment means but it would not substitute front end solutions to end users therefore understanding and collaboration would need to be up front and central which is why the ECB has launched public consultation on the digital euro report which will run until the 12th of january next year as Ulrich also mentioned so this is just a quick overview of the the different aspects that were covered in the conference this morning i'd like to thank all our panelists and presenters and speakers for guiding us through today all of the participants in the audience who were very active with their questions unfortunately as Aline said we didn't have time for all of them but we will definitely be following up and we look forward to having such sessions again thank you very much thank you Fiona this is the end of the conference today so i just want to remind all of you that the recordings of today will be uploaded on the ECB website after the event so in case you are not able to watch the whole session you'll be able to view it here we also provide you with how you can follow us and stay tuned on new events or other news from the ECB thank you for my side as well thank you everyone


DRD, Differential Reinforcement of Diminishing Rates of Behavior, which I don't know why isn't a DRD ROB. But anyway, differentially reinforcing diminishing rates. So if I start out speaking like this, it's speaking very fast, right? So you could then start to reinforce me as I start to speak a little bit slower. Then if you still don't think that I'm speaking slow enough, you could reinforce me when I start speaking at a rate about like this. And now that I'm speaking at a rate about like this, you could reinforce me if I start to speak a little bit slower. Now we are reinforcing a diminishing rate of behavior. As my rate is decreasing, then you are delivering reinforcers. At some point, my rate might get a little too slow. So you might have to engage in an alternative behavior in order to reinforce me back up to a particular level. But anyway, the point is that you differentially reinforce the organism as the behavior rate lowers. That's it. Diminishing rates, diminishing, decreasing, drifting, depleting.


I'm Deborah Gordon-Taslow. I've lived in the Rogue Valley for over, gosh, I got here in the beginning of 1972, so almost 50 years. When I arrived here, I was 20, I'm almost 70 now, so I arrived with my first husband. I had married very young, and feminism was a new thing to me. When I had gotten married in 1970, I had figured out that I would learn how to cook and make dinner every night from husband, etc., etc., and that quickly devolved. I had an older sister who was a little bit of a revolutionary and a college graduate by then, and she was sort of schooling me in feminism, which was a new thing that Mark, my ex-sister, and I would have to share the chores. I remember when it was his turn to do the dishes, I would let those dishes pile up until he did them for days and say nothing, and then it was my turn after he did it, so I was schooling him in that. So I got involved in the early feminist movement in southern Oregon by way of the Women's Health Center, and I'd gotten here in 1972. So in 1973, I went to a benefit garage sale for the Ashland Women's Health Center that was held down in the basement of, I think it was called the Pillars or the Owl Coffee House. It's kind of where the Intaj Indian restaurant is now, but underneath. And I walked in, and there was Amy Horowitz in a pair of roller skates, and she was flitting around, and she saw me. I was wearing an Israeli embroidered top, and she had been to Israel a few times, and she skated over to me, and that was the beginning of my involvement with the Women's Health Center, which was, for the first time, we had an off sheet. Amy was the VISTA volunteer. She had somehow arranged to get paid to organize the Women's Health Center, and so we had offices that were up above, not the plaza, but where it used to be the Ashland Steakhouse, which was one of the only restaurants in Ashland, the Steakhouse Cafe.  So I'm saying we, because it was really Amy who had organized it, and then it was kind of a collective of women, but she was the paid organizer. I think, you know, what was amazing about that time was, I remember thinking, there are all these women here who have not ever seen parts of their bodies, like we have gone to gynecologists who are men, and we have been told how to take care of our bodies, how to have babies, how to, all of that. And we were now, even though we were not medical professionals, we were saying, oh, these are our bodies, and we're going to find out. We would actually climb onto these, we had exam tables, and I remember we would have like training sessions. I remember volunteering to be the model, and getting up on the table, and there's all these women with their, you know, with a speculum and flashlights, like giving exams to me. I would never do that now, but it was so exciting to have this happening that we were learning, oh, with a mirror, this is what this thing inside me, my cervix looks like. We had no idea, and Amy was at the helm, and we would have meetings, just these meetings were the first time that I had ever had meetings with women, you know, that were all women, and they were lesbian women, straight women, older women, not, I don't remember anybody, to me, old was 40 then, right? But there were women who were in there, Amy and I were in our 20s, and there were a lot of women who were in their 20s and 30s, and then there were some in their 40s and so on. But we would all talk not only about the organization, but what we were going to do, but talk about our lives, and that in itself was revolutionary. Some of us were married, most were not, I had gotten married pretty young. So, you know, we would, I remember talking, we would talk about sexuality, we would talk about abortion, I was mentioning to you later that the health department, the Jackson County Health Department was involved through Amy, and the health department director Al Kearns would come and talk to us about how to do what to do, and I can't imagine that he would give over all of this stuff to lay professionals, but he was very supportive, I mean he was genuinely supportive, not just because the health department had directed him to, and I have to say that Amy has a lesbian, had a way of attracting men, and there were a lot of, she has a lot, and continues to have a lot of male supporters who are very into helping whatever Amy wants, because she's magnetic. And so he would kind of train us in what to do. I don't recall being involved in doing a lot of the medical stuff on clients who came in, I was mostly kind of helping with the organizing, and with grant writing, because Amy and I discovered very quickly that we were simpatico, and one of the ways that we were simpatico was that we were both writers, and we were both articulate. So we would sit down to write a press release or a grant, and she would start a sentence and I would finish it, and we would write grant proposals and press releases, and she would send, I remember her sending me out to talk about the Women's Health Center. She sent me out, there was a women's conference, and this was probably one of the first at Southern Oregon University. It was in the student union, the rogue room, rogue river room, and so I prepared a speech and so on about what the Women's Health Center did, and then I remember like nobody came, and I had this speech, but it was just, you know, there's probably about 10 women there, because either it hadn't been advertised or there wasn't quite enough juice generated yet for women to want to come. One of the things that we did, Amy and I and a couple of other women took a field trip out to Kiliman, where there were communes, and we went to the Magic Forest Farm, because another woman, Jean Sargent, who was a close friend of Amy's at the time, she had lived there and was now living in Ashland, and we went out there to specifically to meet Jim Shames, who is now the director of Jackson County Public Health, but he started, he and his wife-to-be, Heidi, started the Tekelma Health Clinic. He was a doctor and a very young doctor, and we all arrived there, and everyone was walking around with that their clothes on, so we took our clothes off, and we went to the Women's Health, not the Women's Health Clinic, but the Tekelma Health Clinic, and Dr. Shames gave us some instructions on how to handle things and so on. So that was my experience with the Tekelma folks who had come from all over to live this ideal life on a commune, and then their idea of free health care for the community was something that we wanted to connect with and then use some of that model for our clinic. I don't know anything about the displaced home workers, but the women in transition, which happened later in the late 70s, I happened to, after my divorce, I lived with a woman named Derdra Sartorius, who had been part of a commune here and so on, and Derdra, who I'm still kind of in touch with too, who lives on the coast now, was one of the first people involved with WIT, Women in Transition, and of course the big mover and shaker there was Rosemary Dunn-Dalton, which I'm sure you know about, right? You know about Rosemary, and she really created the Dunn House, but this program was based at SOU, and Derdra was one of the women who worked with the WITs, the Women in Transition, and there was a whole group of them who ran a counseling center or a women's center at SOU in the later 70s for many years, so, and I'm not sure what happened with that program, but that there were all these women at SOU or associate, I don't think you had to be a student to come in and use those services, so there was mostly counseling services helping women to, you know, figure out the logistics of their lives and particularly women who needed help or shelter or support in whatever way that was, and there was a whole group of women who were involved with that, but there was this heady sense of taking control of our bodies and our lives. I don't remember if the book Our Bodies, Ourselves, had come out quite yet, but that was a very important book for many of us. Ruth Alexander, who was one of the women on the Boston Women's Health Collective, ended up living here many years later. Our daughters were friends in the 80s. Anyway, 80s and 90s, she still lives here, but she's not well anymore, but anyway, there was this whole sense of we were doing this, and I remember there was a little bit of a disconnect because I was also a college student, and I remember being in a, like, it was required to take, I don't know, communication, speech class in whatever liberal arts curriculum I had, and I remember being in this class and everybody, they were all like 20-year-olds like me, many of whom were male. I remember giving a little speech about feminism and about how I had discovered that my beauty wasn't my only asset and that how intoxicating it was to not have to shave my legs or my armpits or, you know, wear makeup, and I remember the horrified looks of the young men in that class, and I wasn't doing it to be shocking. I was just like saying, wow, this is this thing. They were like, no, I can't be. So, you know, I was living in that world and also the world of women being involved with women's health center. You know, I remember, you know, we would talk about things, we would talk about abortions, we would talk about sex, we would talk about our particular, like we would talk about infections and things like that. And, you know, I remember one time I was getting, having, going to get an IUD inserted. I remember Amy and my husband came with me, but it was very much in contrast to when I first had an IUD, I was having one taken out and another one put in because I had the Dalcon shield, which was determined to be dangerous. But when I had first had it put in, I remember when I was, I don't know, 19 or something back in Los Angeles, going to Kaiser Permanente and saying that I wanted an IUD. And I remember, you know, waiting in the waiting room and the doctor came in and he said, I mean, this was the conversation we had. He said, here are the ones that I've taken out of other women and he threw a bunch of them on the, on the table and said, if you really want this, I'll do it.  I don't want to have any child and et cetera, et cetera. And he said, fine. And he shoved it in. I, I passed out from the pain and the guy was so insensitive and that was a fairly typical response to women's, rather than having a conversation, what can we do about birth control? How do we, you know, whereas now what was happening was when I went this time, Amy came with me, Mark came with me, we, it was like, we're taking care of you and we're going to talk with the doctor about what you want, why, why you want it or don't want it or, but that was a very new thing. You know, my sister had recently had a baby and it was revolutionary that she had Lamaze classes and she didn't have a home birth like I did a few years later, but she had natural childbirth. That was like revolutionary. It's still not everybody does it, but women were, you know, still being sort of, you know, locked into stirrups and drugged and, and the idea was that men were in charge and now men weren't in charge. You know, we were saying we have some say so in our bodies and our lives. I'm trying to remember who else was, was involved, you know, besides Amy and I in the health department. There was a woman named Julie Chapman who was very prominent. Judith Wright Visser still lives here and was very involved. I don't remember. There were women from all over. Cisco Davis who was Jane Davis at the time was involved. There were not too many women that had children of course because they didn't have too much time, but it was kind of a variety of women and a group effort.  That women's health center became the Ashland Community Health Center. I don't know if you know that and Kathy Shaw was one of the, I think she was the executive director and my friend Julie Tidalbaum who was Julie Schwartz then worked as a technician there. I don't know how that happened, how that transition happened.  But I think the whole gestalt of what was happening stayed with me and in that sense, you know, I became in the 80s and so on. I became a storyteller. I became a teacher and a storyteller. I remember my first teaching job in 78. I taught third grade in Central Point and I remember we had career day and on that day we were supposed to talk about careers. And Central Point was not quite as highly conscious as Ashland. So in my, we had to do activities that had to do with careers. So my activity was, I did a survey with third graders and we had different groups rotating in and I listed a bunch of jobs and said, what do you think of male or female when I say these jobs and they would check it off? And then we talked about the stereotypes with these little eight year olds and nine year olds. And I remember the other teachers thinking this isn't a career day activity. This is that Ashland teacher with a feminist. But it was eye-opening, you know, to the kids and then to talk about, well, if I say doctor and you say female, what is, why wouldn't a female be able to be a doctor? And it was really interesting to have these discussions and see what the boy said and what the girl said and so on. So I carried that in and when I became a storyteller, very quickly I realized that all of these stories that I was telling, and I was telling them to kids and now I tell more to adults that all of the heroes, and these were mostly European fairy tales and so on, folk tales, all of the heroes were male. And at the same time, I was realizing that stories have a deep influence on our psyche. I was reading, you know, Carl Jung and Bruno Bettelheim and so I started looking for collections of stories that had women who were more than just passive princesses waiting to be rescued. So that became a thing of mine that I would, I went around after, after I started having kids in the 80s and then became a professional storyteller, I would go and tell stories at schools and stuff and that was my, part of my specialty. So, and it remains that to this day when I tell stories and I tell stories in, my husband now is a rabbi, different husband than the first one. You know, we've been married for about 40 years and my specialty is telling Jewish stories now and spiritual stories, but I kind of, I change those with my storyteller's license. In fact, I'm working now on a collection with another group, with a group of Jewish women who are taking some of the old traditional Hasidic stories and changing the endings or featuring women because even God language, calling God he, all of that and having the main characters be men has a huge influence on girls who are listening to these stories. You know, it's really discouraging to have your role models only be, you know, passive and sort of helpmates rather than active heroines. So, you know, there's still so much work to be done.  You know, countries all over the world have had women leaders, but not the United States. So, I mean, sexism is so incredibly insidious and pervasive, you know. So, you know, I've seen a lot of changes. We've had women mayors here, we have another one now, but women certainly have more of a strong voice than they did in the 70s, but there's still, you know, a deep sense of sexism that has permeated our culture that, you know, that we still have to deal with. Yeah. The women at the Women's Health Center were women who lived here, although, you know, they were, they had come from other places. I remember sitting at a meeting and there were all of us kind of hippie women sitting there and in walked. Oh, what's her name? Annette Pugh and Annette and Lance, who are both dead now. Annette was also a young woman a little older than us, and she was wearing a beautiful dress and walking in very wriggly, and she owned a whole part of town, town she and Lance, and Lithia Grocery was the main one that she owned, and she walked in and said, I would like to help, and we were like, you would.  And she came in and I don't recall her being involved later, but I don't know if she donated money or if she got involved in an activity or whatever, but she was local, but she had, they had moved here from somewhere else. And all of the other women lived here. And including those at the gathering, I don't recall there being other women, but there were plenty of women in Southern Oregon who were interested in coming. So Amy was really responsible for saying, let's have a women's festival. And she, I talked to her recently and we've remained friends for 50 years. I'm the godmother of her daughter and so on, but she says that there was an original meeting of women's space that I don't know about it. But what I remember was having a big, we decided to have a big gathering, and it was what's now called Camp Latgawa, then it was called Soda Springs, and or at least we called it that. And we had several meetings to organize that. I mean, there aren't many people who can organize a big event, especially, I mean, Amy was 21 years old, but it was, there were hundreds of women and we had all kinds of workshops. So this came out of Amy's work with women's health, but it really was the outgrowth of feminism and the women's culture that was evolving. And I remember we had a lot of meetings up in that health center to organize this conference. It was a weekend, it was like a two day, yeah, because there was one overnight. There were workshops on, I remember going to a writing workshop, I remember going to Sylvia Goodman, who was a Rykeian therapist in town, and she had a whole workshop. I remember trying to remember the name of the woman who was in charge of food, but we all brought all kinds of food and she cooked huge meals. I remember spending the night in a cabin. I remember staying up all night talking with other women. I mean, there were so many things that were going on there. And I remember I made a huge banner and I have no idea if this banner still exists that said women's space. And I had gotten the idea from my sister, but I had all these made of fabric women's symbols on this huge banner. And then I made a big circle and then around that I had, and I'm not a much of a stitchery person, but back then I was motivated. So we have this huge banner in fabric that said women's space. Apparently it became women's source after that. That was the only gathering that I went to. And 20 years after that, so that was like in 72 or something or 74. It was before Amy, I must have been about 74. Yeah. 20 years later in 94, I was part of a true kind of an improv group called Playback Theater. And we were asked to come up to the, it was a Girl Scout camp by Lake of the Woods. And they had the 20th reunion of women's, it was called women's source. And we came up and there were about, oh gosh, I don't know, there were, there were maybe, there were less than a hundred women, but it was a good size group in a hall or whatever. And then we did a playback performance of women's stories through the years. A lot of my friends now who were in, you know, 60s and 70s. I mean, I don't mean the 1960s and 1970s. I mean, we're, we're older now. And you know, we, we've mellowed a bit, but we're still, I think, adamant about being in charge of our lives and our bodies. So some things have changed and some things haven't. So you and I were talking a little bit about the issue of right to life before. And I would say that in the 70s, for the first time, you know, we were talking about, you know, abortion had not been legal. And I was saying to you before that in the, in 1968, at 17 years old, I had an abortion on a kitchen table in Sunset Boulevard, which was a horrifying, terrible experience. And when I got to Southern Oregon and in the 70s, and it was the first time we were really talking about abortion, I was talking with women who were having abortions and kind of taking it, not for granted, but very grateful that now we could get safe and legal abortions. That was a huge step. And you know, I'm a rabbi's wife now. And so I'm a, I would say a spiritual religious woman, but I'm adamant about right to life. I'm not right to life, but right to choose. And you know, I was saying to you before that I have a nephew who was a big mucky muck in the narrow organization in New York. And we've had conversations about this because I have what would call a religious view that includes a woman's right to choose, which I think too many fundamentalists is unthinkable. And you know, my belief is that at the moment of conception, there is life, which is a position that's untenable in terms of fighting for abortion rights. But I can reconcile that because I feel like it's part of the incredible abundant life force that is God. And it's available all the time. Life is everywhere in the trees, in the grass, in the birds, and being offered all the time, all the seeds that are dropping from the trees. And I throw out a lot of those seeds. And I feel like a woman has the right to say, thank you, God, for the sacred offer. No, thank you. This is not the time for me to bear life. I think women are the sacred holders of life. And we have to be given absolute sovereignty over our own holy bodies and totally are capable of saying no. And to me, that's not any kind of sin at all. It's just a saying, no, thank you, not right now. And there will still be abundance.  I've been part of doing mikvahs after abortions and miscarriages and so on. Mikvah is a ritual immersion. We have a mikvah bath in Southern Oregon and with women and as part of a lot of women's ceremonial work that I'm involved with as the wife of a rabbi. So I mean, I don't know if you want to include any of that or if that has anything to do with the history of Southern Oregon, but that's kind of how it's manifested in me as a spiritual woman who would still call herself a feminist. You know, my daughter, by the way, who's just turned 39 the other day, is a midwife and she has a PhD in women's studies and she has a birthing clinic in Uganda. She implemented it, started it, funds it, but it's run by the traditional birthing attendants there. And, you know, I like to think that my spirit of feminism inspired her, although she's a pretty much self-made young woman, but, you know, she's working in an area of women's bodies and is a very strong feminist and is very influential in that area in the world. So that's kind of a little effect. And Amy's daughter is a young feminist as well. She's a lot younger, but yeah, I like to think we've spawned this generation of women. That's the only way I can take credit for, you know, probably Rachel, my daughter's done more in her life than I have in my life and I love that. I love the fact that she's out there doing the work. Well, I don't know. I would say that, you know, now almost 50 years later, we're looking at a world that in many ways looks very different. We have a woman vice president. We have women news anchors, etc, etc, etc. But I feel like there's still a lot of work to be done.  There's still a pervasive sense of who women are and women's place. And if you talk to any of these women, they will tell you that they have had to work tooth and nail to get where they are. So there's still a lot of work to be done. I mean, we have right now we have abortion rights, but they're really threatened. And we have the Supreme Court that's come in the new. So I'm banking on the strength of the younger generation, like my daughter who is a feminist and a midwife, like so many of the young women that I know, children of my friends who are doctors and lawyers and are doing amazing things in this world. And you know, I think that we sowed those seeds and we were really lucky to be at the vanguard of, wow, we could actually do things for ourselves. We could take control of our own bodies. We can, we can perhaps direct the conversation. So I hope that that directing of the conversation has manifested in a larger conversation that's still going on, but that has, you know, a little bit of the blossoms from the seeds of what that we sowed in the 70s.


Hello, welcome to NPTEL NOC, an introductory course on point set topology part 2. Today we will have the last section on the dimension theory, module 44, local separation to global separation. Recall that our study of dimension theory began actually in the previous chapter with a discussion of separation properties which we have named S0 to S3. Keeping in mind that we are going to restrict the class of the colloidal spaces for those which are separable and matrizable and in particular T1, we have pointed out earlier that S1, S2 and S3 are some stronger forms of house dwarfness, regularity and normality respectively. We may call this itself the first step in the passage from local to global. Moreover, the passage from SI to S3, S1 to S3, there is one S2 in between, itself can be viewed as a passage from atom to mass, atomic to aggregate or whatever you want to say, mini-school to largerness and so on. A kind of local to global that is also you can say is the passage from local to global. Recall that one of the first few results that we proved in the previous chapter is that Lindelow plus S2 implies S3. So that made S2 our central object of study. Next in this chapter we adopted S2 to represent zero dimensionality and then inductively obtained its higher dimensional versions to define higher dimension manifolds. Then came the theorem 9.13 and 9.15 which you may term as another step toward globalization. Let me just show you these steps. This was the theorem that if we have subspace of matrizable space, separable matrizable space then x as dimension less than or equal to n if and only if given any closed subset C of x and a point outside there is a closed subset D of x such that dimension of D is less than or equal to n minus 1 and x minus D is equal to a separated b and we are both open and closed subset disjoint with p inside a and c inside b. So we have pointed out that in the case of n equal to 0 dimension of D is minus 1 means D is empty set this is precisely the condition S2. So similarly 9.15 we can also similar things I just wanted to show you one of limit here. If x is a matrizable space and x prime is a subspace of that dimension of x prime is less than or equal to n if and only if for every point b inside x now I am putting condition on points of x all together I am getting condition like that. Condition is dimension of x prime is equal to n what is the relation between points inside x and the subspace will dimension less than equal to n. So that step is again another step or globalization this is what we meant okay. So let us go back to what we are doing today. So as the inventor, inventor had termed it the success of the concept of this theory of dimension why I am calling this theory because there are other theories also hinges upon successfully strengthening the passage from local to global. This is the topic of this last section wherein we shall be able to reach our goal the final goal of proving that the topological dimension of the pyridian space rn is actually equal to n. So that I call it as a success of the theory. Here is the next step in the passage from local to global okay.  Given any two disjoint closed subsets c1 and c2 there exist a closed subset b of x separating c1 and c2 such that a intersection b is empty. The c1 and c2 are disjoint closed subsets by normality you can separate them by open set that is a different aspect. Of course that will be the starting point in the proof of this. What we are going to do is there is a separation by a closed subset b it does not intersect a at all okay. So this you can call it as really a crunching of of course we have to improve on this also of you know the separation properties being you know globalized globalization problem here. So let us start the proof of this. We have to produce a closed subset b of a b of x contained in the complement of a such that when you throw away this b x minus b can be written as disjoint union of closed one sets v1 and v2 c1 contained inside v1 c2 contained inside v2 that is the meaning of separation of x minus b. Choose open sets ui i equal to 1 and 2 such that u1 bar intersection u2 bar is empty and cis are inside ui so this step is just merely the metric space property here normal as normal property of a metric space okay. So that is all we are using here. Once you have this u1 and u2 look at their intersection with a in fact take ui bar intersection a these will be disjoint subsets of a and a is of dimension 0 so apply the s2 property there okay. We have a equal to c1 prime separation c2 prime here ui bar intersection a is inside c i prime okay so that is the definition for that is the property for the 0 dimensional state. Now you enlarge the c1 prime c2 prime along with c i say put f i equal to c i union c i prime i equal to 1 and 2 okay. Look at these c1s are subsets of a c1 c i prime c1 c2 are subsets of the largest space x okay cis are closed c is are closed inside a but a is that is not closed inside x so there is some problem here otherwise f pi 2 would have been easily you know you can take it as closed subset you are take the a as a closed subset no here this dimension 0 that is the point here so we shall show that f1 and f2 are mutually separated sets which is stronger than saying just disjoint closed subsets okay so I mean they are not closed subsets but f1 bar separates it does not intersect f2 and f2 bar does not intersect f1 so mutually separated subsets so which is slightly weaker hypothesis than having disjoint closed subset disjoint closed subsets are easy to separate so this is into more so for here we will need more than normality namely complete normality wait a minute so let us have this one now how to prove that this one namely f1 bar intersection f2 is empty indeed once you prove this one the other one is symmetrical all these things are conditions are symmetrical in the other one so that will also prove exactly same okay so once you have that use complete normality of the metric space it follows that we get open subset w in x such that f1 is inside w and closure of w does not intersect f2 this way it is easier to state f1 contained inside w closure of w does not intersect f2 you can then take b as the boundary of w automatically boundary of w is closed so this b is a closed subset which will separate f1 and f2 because f1 will be contained inside w does not intersect b f2 does not intersect b because it does not intersect w bar at all so f1 and f2 are inside that one if you take w union complement of w bar that will be precisely equal to the whole x minus boundary of everything x minus b okay so moreover if you look yeah that is what I have told b intersection a okay is empty because why I just why this is this I have not yet shown what I have shown f1 and f2 are different moreover you want to ensure that b intersection a is empty but b intersection a b is boundary of w boundary of w intersection f1 union boundary of w intersection f2 okay because this entire a is union of c1 prime and c2 prime and f1 and f2 contains even prime c2 prime so a is contained in the f1 and f2 therefore this intersection is intersection with f1 union intersection with f2 but both of them are empty okay just now we show them so this b will serve the purpose all right so we have yet to find out that one so here is the schematic picture of what is happening started with c1 and c2 which are disjoint closed subsets these ellipses then enclose them in open subsets u1 and u2 that closure of u1 and closure of u2 are disjoint intersected u1 with a this is my say a is of dimension 0 that is why I have shown like dot here okay intersect that with a1 so there you get some subset here like this okay up to u1 here similarly from here to here all right so they are disjoint inside a and they are disjoint closed subsets so you can separate them by c1 prime and c2 prime so c1 prime and c2 prime may go out of u1 they may be out of this one so here it might come out of u2 also here to here and so on so of course this dot dot dot for dot dot dot ellipses indicate that after all an open subset of subset of a is nothing but some open subset inside x intersection with a so that is what I have shown here then I put f1 equal to c1 union this c1 prime up to here c2 in c2 prime up to here I have to show that they are mutually separated in the picture it is obvious you can't use the picture to prove a theorem okay you can take the help but finally everything should be purely logical and finally what we want is this green thing w such that it is closure okay does not intersect this f2 c1 prime union c2 prime and this boundary if you take boundary if you throw away that you get a separation all right so let us prove first of all that these two namely c1 union c1 prime c2 in c2 prime they are disjoint and mutually separated is what I have shown of injection so let us do that so it remains to prove 31 31 is that this f1 bar intersection f2 is empty okay the other one is similar so by symmetry it is enough to prove that f1 bar intersection f2 is empty first of all f1 bar f1 itself is c1 union c1 prime therefore f1 bar bar denotes now everything denotes happening inside x okay closures are inside the whole space x f1 bar is c1 bar union c c1 prime bar because it is just a union of you know finite union but c1 is already closed so it is c1 union c1 prime bar therefore it is enough to check that c1 intersection f2 okay this is c1 intersection f2 and c1 prime intersection f2 they are empty then f1 bar intersection f2 will be empty so this is the first step I have to show these two things now just to show one of them I have to show these two things okay so let us see now c1 is inside u1 by the very choice u1 u1 is an open subset containing c1 hence c1 intersection a is you contained in u1 intersection a contained in u1 bar intersection a but u1 bar intersection a c1 prime okay and hence c1 intersection c2 prime is contained inside c1 prime intersection c2 prime okay just now c1 intersection a is already in c1 prime c1 intersecting c2 c2 prime is inside the subset of a so I can take intersection with a itself so that is contained in c1 prime intersection c2 prime and that is empty to begin with okay so instead of c1 it is in f2 I have looked at c1 intersection c2 f2 has two two parts right what is c2 and c2 prime the c2 prime okay so c1 intersection what we do here c1 intersection f2 will be c1 intersection c2 which is empty c1 intersection c2 prime okay so this intersection is now c1 intersection c2 prime now what you have what is happening to this one so c1 intersection c2 prime is again a intersection c1 intersection c2 prime everything happening is c2 prime a intersection u1 bar intersection c2 prime because c1 is contained inside u1 bar okay so but that is contained in c1 prime intersection u prime so that is empty okay so I am more or less repeating this one here all right next to show that c1 prime intersection f2 is empty okay so how do you show there are two parts f1 bar has two parts right one we one we showed the second part is this one this one I have to show for each x belonging to f2 we shall produce a neighborhood of x which does not meet c1 prime then it follows that that point is not in the closure of c1 prime because a whole neighborhood does not interact that if x is inside c2 there are two parts to f2 one is c2 and another is c2 prime if c2 you can just take v equal to whole of this u2 for all the points inside c2 take just v because v which is equal to u2 here is contains the whole of c2 okay and then v u2 is not a intersection intersection c1 prime is empty right so u2 intersection a which is contained inside u2 bar intersection a that is contained inside c2 prime and then u2 intersection c1 prime is u2 intersection a intersection c1 prime which is contained in c2 prime intersection c1 prime that is empty so method is the same but argument is different that part takes one part if x is in c2 prime then what do I do but c2 prime is an open subset of a it is actually closed subset right because separation okay c2 prime is also open inside a so we get an open set v this time it is a different v I have to do okay so this v is inside x open such that v intersection a is this c2 prime open subset of a then what happen v intersection c1 prime is v intersection a intersection c1 prime that is c2 prime intersection c1 prime okay that is contained inside so that is also okay so separately we have shown that c1 prime intersection f2 as well as c1 intersection f2 are empty it just means that f1 bar intersection f2 is empty similarly f1 intersection f2 bar is also empty and that completes the proof now we can state a more pliable statement and easy to remember statement start with any separable matrix space take a subset of dimension less than or equal to n where n itself is finite of course I assume bigger than or equal to 0 because if a is empty there is no statement that those things we have seen already so given any two disjoint closed subsets c1 and c2 inside a there exists a closed subset v of x separating c1 and c2 such that the subset a intersection v okay has dimension less than or equal to n minus 1 okay so from 0 dimension we have come to arbitrary dimension here now okay so how do you do that of course x is a matrix space there exist open subsets u and u2 such that c i's are inside ui i equal to 1 and 2 and intersection u and u2 is empty this is normality because c1 and c2 are disjoint closed subsets now suppose n is 0 this n is between 0 and infinity right so n is 0 there are two cases to be handled if a is empty in which case we can take b equal to u and u and u2 complement that's a closed subset and x minus b is just disjoint union of u and u2 over okay that's what when this is 0 otherwise dimension of a is 0 it cannot minus 1 okay in which case the first earlier lemma which we just now that we give you the required result so we have the inductive hypothesis here now suppose n is bigger than 0 using a previous corollary we can write a as union of two subsets d and e where dimension of d is less than or equal to n minus 1 and dimension of e is less than or equal to 0 this is one of the some theorem that we have derived last time right now you use the above lemma to give a closed subset b which separates c1 and c2 such that b intersection e is empty I don't know what is happening to d we'll come to that later but e part is empty that is the the starting point with n equal to 0 but then this implies you take a intersection b that will be now contained inside d but d is of dimension n minus 1 less than or n minus 1 so intersection of a a intersection b is also of dimension less than or n minus 1 okay so this after hard work of lemma this comes quite easily by of course I have to use this crucial thing here namely anything which is of dimension n can be written as union of two subsets one is dimension minus 1 another one is 0 now in the above theorem take a equal to well this was not a theorem it is a proposition it does not matter take a equal to x then what do I get that x be of dimension less than equal to n then any two disjoint closed subsets can be separated by a closed subset of dimension less than or equal to n minus 1 okay so no question of intersecting with a because a is the whole space now we want to improve upon that one let x be of dimension less than equal to n and c1 c2 c3 and similarly this even prime c2 prime etc pairwise disjoint closed subsets c i intersection c i prime is empty that is the meaning of that how many are there n plus 1 this dimension less than equal to n there are n plus 1 pairs of disjoint closed subsets then there exists a closed subset bi there exists closed subsets bi how many n plus 1 of them such that each bi separates ci and ci prime and intersection of bi i rank 1 to n plus 1 is empty all right so how do you get this one it is also easy from previous theorem apply to c1 and c1 prime you get a closed set b1 which separates c1 and c1 prime such that dimension of b1 is less than into n minus 1 now use the proposition or the theorem we get a closed subset of of x a subset b2 separating c2 and c2 prime such that when you intersect it with b1 it is dimension less than to n minus 2 because you already n minus 1 now you keep on doing this you know repeat this step get a b3 such that it separates c3 and c3 prime with dimension of b1 intersection b2 to n minus 3 how far you can go till you get minus 1 and that is empty so you have to have dimension less than equal to n here and there must be n plus 1 of them then only you will actually have all right now we are very close to the end here okay the rectangle or rectangular box minus 1 to plus 1 interval raised to n contained inside rn okay suppose you have c i plus and c i minus denote its faces defined by equation x i equal to plus minus 1 if you just n equal to 1 this is nothing but c i minus 1 and c i minus c i plus is plus 1 that is all if n equal to 2 there will be four faces pair of two of two pairs of opposite faces right so that is the way you have to take these faces defined by the equation the coordinate x i equal to plus minus 1 okay okay now for 1 less than or i less than n there are n pairs here suppose b i is a close subset of jn bar which separates the opposite faces c i plus and c i minus it just means that when you throw away b i from j n bar you get two open subset each of them containing c i plus or c i minus one of them contains the i okay that is separation suppose you have got these b i like this then we want to show that intersection of b i is non-empty the crucial thing here is what there are only n of them the previous theorem says if there are n plus one of such things then it is empty so together they are going to imply a big theorem for us however the proof of this is now based on something different that we did last time namely Brauer 6 point theorem comes here okay let us see how pay attention to the method of proof because that can be used in several other places okay let did you know the Euclidean distance function in R n for each x in u i plus I have overly told you what are u i plus here u i plus is what the open subset containing c i plus and this one containing c i minus u i minus c i minus okay for x belong to u i plus let p i be the root to be the foot of the perpendicular from x to c i minus so this is a one of the faces so go all the way to c i minus from u i plus don't change the ith coordinate that's all the p i and x have same ith coordinate it is the foot of the perpendicular from x to this plane then this is now elementary observation distance between x and b i b i is a subset which separates the two things right it's less than equal to distance between x and p i and this distance is actually put distance between x and c i minus why because p i is the foot of the perpendicular and what is this distance is just one plus the ith coordinate of x okay similarly so what we have proved by this one is distance between x and b i is one plus x i distance between you know x and b i is less than equal to one minus x i for every x inside u i minus this is for u i plus this one u i minus very easy to remember let me justify this one with a small picture here n equal to 2 okay so this is minus 1 to plus 1 minus 1 to plus 1 the product square okay this is square and this is c 1 plus x 1 coordinate this is c 1 minus this is c 2 minus and this is c 2 plus this is my b 2 separate c 1 minus and c 2 minus take a point in u 2 plus this x take its projection on to this axis on to this plane so here is just an axis what is it it is x 1 coordinate will not change what is its y coordinate y coordinate of this point or x 2 coordinate of this point is nothing but this x 2 coordinate here plus this one the distance sorry the distance is this distance is 1 this distance is x 2 right so distance between x and this one is same thing as distance between x and this part which is bigger than distance between x and b 2 and that is precisely what we have to do this x b i less than 2 x b i is a dx c i minus now i define a function here for one less than or less than equal to n define a phi from j and bar to r as follows if x is in u i plus just take it as distance between x and b i that is a continuous function right we know that what is this this is the minimum of distance between x and little b i where little b i runs over b i if x is in u i minus you put a minus sign there so x c is in b i put a 0 look at this one if x is in u i bar this this u i plus it is this one if it tends to a point you know if you take limit tends to a point inside the boundary that this distance becomes 0 right similarly this one also this will become 0 so by these are continuous this f i will be continuous because of that okay now you combine these two in equations here inequality what you conclude is that minus one is always less than x i minus f i of x is always less than good one there are two different cases you may have to work you have to work out is very easy okay you have to just use this and this one according to this because there are different definitions are different the function f i if different you define that's why okay so x i minus f i is between minus and plus one therefore if i define okay f x equal to f 1, f 2, f n and g x equal to x minus f x then what happens this is a function taking the values inside r n but g x will be taking function inside j n bar it will be always between minus and plus one all the coordinates so obviously both of them are continuous okay therefore you have got a function from the closed rectangle j n bar to j n bar you can apply braver six point theorem so we get a point x is that g x equal to x what of that mean f x equal to 0 what of that mean what of that mean f i of x equal to 0 for all i what of that mean x must be inside each of this d i which means x is in the intersection of here okay so that is the statement here okay this theorem is proved now as i told you now you can combine this with the previous theorem you get a wonderful result now tie it thing namely dimension of j n bar has to be equal to n okay let me go through this one it is not so clear if not dimension is always less than equal to n because because of what we know that part we have already proved okay dimension far in less than equal to n and this is a subspace so that part we have already proved so dimension must be less than equal to n minus one as soon as it led minus one by our theorem 9.38 whatever we have proved today it means that so for each i 1 less than 1 less than 1 less than 1 there exists n closed subset b i which separates c i plus and c i minus such that the intersection is empty but braver six point theorem applied now just now the previous theorem says this is unempty so that the contradiction to this theorem right 9.39 therefore dimension is right it has to be equal to n r n being a larger subspace of the you know containing j n or j n bar already so this dimension is also n alright so we have proved that not only this one can you can take now any open subset any take any open subset which is homeomorphic to some d n j n bar and so on contains something etc right any open subset will contain some some copy of this one therefore all open subsets will be of dimension n inside r n all non-empty open subsets of course so here we remark topological dimension whatever you have defined okay is invariant for the class of separable matrizable spaces we have not defined it for arbitrary spaces that is one point you have to remember thus we may also conclude that you know the browser invariance of domain which is a weaker form of this one weaker form of b i d so I am going to give that namely r n and r m if n is not equal to m cannot be homeomorphic because we have just shown that dimension of r n is n dimension of m but dimension is a homeomorphism invariant okay this is weaker form of browser invariance domain of course the the real browser's invariance of domain is the following which is which is a which is a stronger form of this one namely take any two non-empty open subsets non-empty subsets of r n suppose they are homeomorphic then if one of them is open the other one is also open and that is why the name invariance of domain the word domain was used more often than just open subsets in the older days so being open is the same thing as being a domain and that is an invariance so that is why it is called invariance of domain okay unfortunately we are not able to touch this one we have come very close but there is still a big gap here okay so the proof of this will take us much deeper into dimension theory which is beyond the scope of this course original proof due to browser uses another topological dimension theory namely Lebesgue covering dimension in modern times it is fashionable to prove this using homology theory so there are many proofs of this work this great theorem for a proof using only simple shell approximation which is actually a part of it is there in for a visual implicitly you may see my book so here is a easy exercise for you deduce 9.42 from 9.43 namely this I said is a this stronger form I said why so you assume this and prove this one okay we have proved it using the whole thing using the all our you know first two different chapters that we have studied so carefully but assuming this you prove this one that is your exercise okay all right so next time we will start a new topic thank you