 So, today we shall take up one of the most salient features of the coherent topology which allows us to practical way of constructing continuous functions. As a consequence we shall prove the existence of partition affinity on series of the complexes. This will essentially prove that they are para-compact because if you have a host of space then para-compactness is equivalent of existence of partition affinity. Of course, the theme of constructing functions by step by step namely skeleton by skeleton will occur again and again in other contexts also. The only thing that we assume is that you are familiar with partition affinity especially if the space is compact and host of then you must be knowing how to construct partition affinity. Nevertheless, whatever is required here I will recall them, I will recall all the definitions and so on. To start with the topological space, a partition affinity on X we mean family of continuous functions. That continuity is important here. Some family indexed by some set lambda say so these functions theta they are all defined on the whole of X taking values in the closed interval 0, 1 they will satisfy one of those two conditions. For each X you can find a neighborhood of X in X such that that neighborhood will intersect only finitely many supports of theta alpha which is same thing as saying that only for finitely many alpha theta alphas will be non-zero on an X. It is called local finiteness of the family theta alpha. The second condition makes it possible, the first condition makes it possible to put the second condition. The second condition says that the sum of all theta alpha at any given point is equal to 1 and that should be true for all X inside X. In any given X summation of all theta alpha must be equal to 1. Some makes sense because of condition 1 essentially because though the family may be infinite the sum will be finite at every point. So that is the definition of partition of X. But now it has to do with many relations with other things namely starting with a covering open covering of X a family theta alpha is said to be subordinate to you the covering. If you have a function on the indexing sets, see lambda is the indexing set of theta alpha and I is the indexing set for the covering. So function should be from lambda twice such that support of theta alpha where alpha is inside lambda which is by definition closure of all the points wherein theta alpha of X is not equal to 0. That is a subspace of closed subset of X that must be concerned inside u of something and that something is beta alpha. So that beta is a function. That function beta is called refinement function. So each support is contained inside some open subset of this cover corresponding this cover. It is a standard result over any Euclidean space any subspace of Euclidean space. There is always a partition of infinity subordinate to any given cover. This is much easier to prove than proving such a thing on a Paragom practice space is whatever they are. Similar to our earlier lemma 1.2 of extending neighbor roots and functions. The key result that we need is to if the following was relative version of union partition of infinity so that functions can be patched up using the patching a process in a CW complex. So what we need is a relative version which means if you have some partition of infinity on a part of a space there must be something extending that to the whole space. So that is what I am going to define here and these are temporary definitions you may not find them in literature elsewhere. So start with a family now I am trying eta alpha here a family of real wide functions 1x. An open cover W of x is said to ensure local finiteness so we want to bring that local finiteness condition more active here condition 1 more explicit. So it is supposed to ensure local finiteness of theta if for each W in W the set of all alpha in lambda such that W intersection support of eta alpha is not equal to empty set this is finite. See local finiteness ensures there is some such cover so if W is such a cover open cover then you say that W ensures the finiteness for theta. So that is just the reference so this is followed by point 1 there and so I am making this definition here so that we can keep talking about what is the cover which will ensure the finiteness. And definition is take y contending z x a subspace now you have an open cover for x and a partition affinity on y subordinate to restricted cover the restricted cover is defined by taking any member of U and intersecting with y so collect all of them that will be obviously cover for y and it is an open cover for y. So for the best cover should be subordinate should be the dominant namely the partition affinity must be subordinate to that ok and correspondingly let beta be the refinement function and let W be an open cover for y which ensures local finiteness so all these things play important role in the definition of extension. Now let us have theta hat ok this is the definition theta hat a collection of psi alpha where alphas are inside another another indexing set lambda prime ok suppose this is a partition affinity subordinate to you such that such that what these are the conditions so it is going to be quite lengthy definition first of all these indexing set here the new indexing set must be the larger than the original intercept ok this containment may be equality also is allowed that is not a problem. For each alpha in lambda for the old members this psi alpha restricted to y must be eta alpha so that is the meaning of psi alpha extensions ok psi alpha restricted to y is eta alpha. And for each W inside W the ensuring local finiteness there is an open subset VW of X so the extensions of now the these open curves themselves VW of X such that VW intersection Y is W so each W is extended to an open subset of of X now and what is the property of this extension the extension set of all the the FW I have defined set of all alpha belong to lambda prime which support of psi alpha intersection with W is non-empty ok so this was remember this FW was all alpha belong to lambda such a support of psi alpha intersection W was non-empty now I want this FW equal to this larger set look at all psi alpha which may intersect W VW the extended thing ok no new member should come there that is the meaning of this FW is equal to this one FW is already inside this one but I want FW is equal to this right hand side ok which just means that you are adding extra members here in the indexing set those functions ok support of those functions should not enter the the open subset so coming from W at all old W's they should be away from them you should not enter even an extensions of those so small extensions must be there so this is a very strong condition ok you have to understand this one carefully ok the fourth condition is there is a refinement function beta prime from lambda prime 2I for theta hat and this beta prime must be an extension of beta that means for old members lambda lambda elements inside lambda beta prime must be beta itself so with all these the indexing set indexing function the definement function is extended indexing set is extended functions are extended ok. And the third condition is that the new functions are somewhat away from the old functions that is something important here ok we then call psi alpha an extension of theta alpha alpha belong to lambda it is a technical definition so that instead of repeating all these four conditions I have just say this is an extension of theta alpha ok thus an extended portion of unity consists of extensions of all old functions to the larger space together with some extra functions to cover up the rest of the larger space it is important that we make the technical assumption that the support of these extra functions do not enter into a neighborhood of the smaller space this is ensured by extending the cover which ensures the local finiteness finally we shall also need to extend the refinement function the extra portion is needed when we want to build up the partition affinity in the inductive process ok all these four points here are important to keep in mind. So here is an elaborate proposition which will ensure the step by step extensions that x be obtained from y by attaching k cells ok so you can put an indexing set for these k cells also j and j that u be an open covering for x theta be a partition affinity indexed by lambda on y ok and subordinate to this restricted covering then there is a partition affinity theta hat which I am indexing by lambda lambda prime on x which is an extension of eta alpha and it is subordinate to cover you ok the statement is over here but we want to say what is happening to all these local covering which ensures local finiteness also ok moreover given open neighborhoods w y y belong to y such that this w y y in y of y is an open cover ensures local finiteness of theta you start with this also hypothesis here you start with instead of just eta alpha and this one you you fix this also then there are open neighborhoods w x hat of x such that the open cover w x hats of x ensure the local finiteness for theta hat but these w x hats are nothing but w whenever this x is inside y w x hat which is w y hat intersection y is w y this is for points inside y so even this will happen ok while extending we can keep track of that this also happens this is the statement of the presentation ok now why you need such a thing that is more curious to you than than the proof of this one I would ensure sir therefore let me go to the proof of the partition affinity using this one first demonstrated that then you will know the role of these these extra assumptions and so on so that later on you can figure out how to prove that ok so assuming this proposition let us first prove the main theorem ok so what is the main theorem main theorem is start with any open covering of a c w complex sense ok then there is a partition affinity on x which is subordinate to you the final real theorems are always neatly stated as briefly as possible ok with all the hypothesis included that is that should be a style of the final theorem so that it is quotable without all the paraphernalia of the the notations that you might have introduced in the proof ok so take an open covering for a c w complex there is always a partition affinity subordinate to that open cover that is the statement ok now how do you begin we start with the zero cells the zero skeleton what is a zero skeleton it is a discrete set ok that discrete set itself will be used as indexing set for the partition affinity now ok so put lambda naught is going to be at the indexing set equal to x naught namely the set of all the vertices of x naught the zero skeleton define these two functions have now indexing the first zero corresponds to the zero level x corresponds to the indexing level the indexing set from x zero to i by the formula p zero x of y is equal to 1 if y is x that means these are indicating indicator functions they are delta functions if x equal to y it is 1 otherwise it is 0 so at each point you take the function identically 1 at that point and 0 elsewhere so that is the first stage construction obviously when you take the sum of all these what happens you get only at any point you get only that function and that function is 1 everywhere else it is 0 so sum total will be all other things are 0 sum total is 1 so it is automatically a partition affinity and since x naught is discrete each point is an open subset here so it is a closest also so support is contained inside there itself right so each x naught by the way is contained inside some number of of u because u is a covering for the whole affairs so it will be subordinate to you also so you shall fix up even the subordinated function here since x naught is discrete this is clearly a partition affinity on x naught put w naught equal to singleton x x naught belong to x so that will ensure local finiteness because there is the only only p 0 x will be non-zero here all other things will be 0 so clearly dubbing naught is an open cover which ensures local finiteness of p 0 x so it is an open cover for x naught okay so it is actually all the singleton 0s are taken singleton points so let beta naught this is the refinement function lambda naught to i be a refinement function I have to choose for each point x belonging to x naught some member of u and if that is i I can write with beta naught of that beta naught of that x goes to that point that i whatever you can choose any so you have freedom here okay inductively now immediately we put the induction step and the previous proposition comes into picture so what is the induction inductive step is precisely we have defined correctly here okay each x n is obtained by attaching n plus 1 cells to x n minus n n cells to x n minus 1 right applying the previous proposition we get a sequence to begins we have constructed already the lambda or not so that will get extended to lambda 1 so now we have partition affinity on lambda 1 that will extend to lambda 2 and so on so you get a tower of what indexing sets they may be all equal I do not care but one is contained in the other is all that I need we have get a sequence of indexing sets that is the first thing for each n n is fixed set x n a partition affinity which I will denote by p n n but it is indexed by the indexing set lambda n now so p n alpha double indexing okay alpha ring of x lambda n n is fixed here and these are partition affinity on x x n this nth skeleton which is an extension of p n minus 1 n minus 1 state p n minus 1 half alphas of course these alphas will will range over lambda n minus 1 okay it is an extension of that as we have seen namely whenever we take an alpha inside lambda n minus 1 restricted to p n alpha take p n alpha restricted to x n minus 1 it must be equal to p n minus 1 of alpha that is for each n a family w n which ensures local finiteness for w n we shall index w n x x belong to x n of neighborhoods of x in x n which ensures local finiteness for p n alpha such that whenever x is in the lower skeleton we must have w n x intersection x n minus 1 equals w n minus 1 of x okay so this was the part of the proposition so we have all these things here okay the fourth condition is for each n the refinement function is also an extension of it so I am just repeating the the things that is ensured by the previous proposition here okay now what we do we put lambda equal to union of all these lambdas you see increasing union of indexing cells for each alpha inside lambda I can choose a k such that alpha first time appears in lambda k that means it is not in lambda k minus 1 or the previous ones the first time it appears in lambda k okay and that k is unique define p alpha okay from x to for each alpha I have defined function right x to i now by the property that p alpha restricted to any n skeleton is p n alpha restricted to n if I knew that these restrictions okay are continuous first of all secondly that they agree see when you restrict this one suppose you take x n plus 1 x n plus 1 the function restricted to x n must be the whole function then only it makes sense that is what is guaranteed by this condition okay that they are extensions right condition one says they are extensions of the whole thing right so once alpha is inside k right so that is the first time some p n alpha will appear there after that it is getting extended only so this makes sense from popular tool it follows that each p alpha is well defined and continuous on x okay that is the property continuity follows because because of what the fundamental property of the coindist apology on the seed of the complexes okay we claim that this theta alpha is the required partition of infinity subordinate that will complete the proof okay and according to you means I have to define a function beta which is a refinement function to take beta from lambda to i to be such that beta restricted to lambda n is beta n for all n remember if I restricted to beta lambda n minus 1 it is beta n minus 1 because beta and restricted to lambda n minus 1 also beta n so this makes sense for all n okay so this makes sense and beta is very defined suppose now alpha is in lambda choose n n such that alpha is in the first lambda n so like here okay so alpha is in lambda n but not in lambda n minus 1 then for all m bigger than n what we have support of p alpha intersection with x m which is the largest skeleton which is simply the support of p m alpha when you interested when you restricted to it is simply a p m alpha by the very definition here okay p alpha restricted to x n will be p m alpha okay support of p m alpha and that is contained inside u beta m of alpha by the very definition of beta m but beta m on alpha is beta alpha because because alpha is in x m once alpha is in lambda n it is also in lambda m bigger than equal to x u beta therefore support of this one successfully is containing u beta so support of support of p alpha itself will be contains a u of beta so is that just beta therefore beta is a defined function for theta and hence theta is subordinate to u so here I am assuming that once this is there the support of this is you know it doesn't go on keeping going increasing it is equal to this one so this is very important okay so x m its larger larger than that once m bigger than n this will be always the support support p m alpha and that we are keeping extending the same alpha for all of them therefore beta is refinement function for the entire of theta subordinate so which gives you subordinate into u now for any x in x say again x is in x k minus x a minus 1 that such a k is unique okay here we define w x to be union n greater or equal to k all w n of x starting with x inside k you get first w k that is the first time you get then you take w n x and so k plus 1 x k plus 2 x k plus 3 x and put union of all of them okay they are successive extensions of the previous ones follows that if you take this w x x in x this is an open cover for x y is an open set intersect with any n it will be w n x and w n x is open inside x n all right of course below x below k it is empty so the empty is fine so this is an open cover for x okay now we claim that this ensures local finiteness for theta okay after that taking the sum it is equal to 1 that part is very easy so let us see how local finiteness comes for f of w f a w x so w x is by definition this one okay this will be union of all the f of w n x and greater or equal to k because these are indexing sets at points here they will have to be in one of the indexing sets on this side at conversely so you see union of all this n greater or equal to k is the starting point k is the starting point so it is n equal to k it will be k no f of w x k for each of them is finite by our inductive step okay and for each n bigger than equal to k what happens the view of f of w n x is equal to f of w n percent and this is the condition that we have ensured here left hand side here there was that this point this condition appears there okay so new entries from n plus 1 n plus 2 etc lambda n plus lambda n plus 2 etc they do not come inside that is why so this is equal to w f of x n plus 1 for this one and this is the same thing as this one successfully keep going on like this therefore f of w x is equal to f of w x k only the k thing with all other things are same as this one which so though it is the infinite union the same set the beginning set is f of w k so it is finite okay that is the meaning of that that this cover ensures the local finiteness finally now again x is inside x k minus x k minus 1 as before after all every x is inside some x k k k k k may be different but that is a unique k like summation p alpha over alpha is summation p of alpha alpha is only lambda k i can beyond those things whatever new things come say for lambda k plus 1 you know that the new functions are not actually 0 on x k okay they are they are not entering so this summation is only p alpha where alpha is inside lambda k i do not have to take a full lambda at all but then this is same thing as p k of alpha because x is inside x k so each p alpha is p k of alpha okay the sum total is equal to 1 because this is a partition of infinity on x k so that completes the theorem so next time we will take care of this proposition okay this proposition will be proved in a two steps okay thank you