 Welcome to module 42 of point set topology part 1. So today we will take up study of properties of topological properties. So recall that we are defined a topological property by which something is true for one space then it must be true for all spaces which are homomorphic. Such a property is called topological property or topological invariant. Then we made a definition this also we I am recalling that a property is called small property if whenever a topology on a given set X has it then all topologies top prime smaller than that should also have the property. Similarly a largeness property is one whenever tau has it and tau prime is a larger topology then tau prime should also have it. Most of the topologies that we have that properties that we have studied in this chapter they belong to the first category namely those which are smallness properties except perhaps the first and second countability they are not exactly of this nature I have already told you. The next chapter will consider those which are likely to be called as largeness properties some of them just like host or some of like first and second countability there will be some which are not exactly largeness properties in our definition. At present what all things they have studied like path connectivity connectivity and all these things right. So what we would like to do is look at a topological property for a space we did be automatically true for a subspace will it be true for a quotient space if it is true for several of them then will it be true for the product space so these are the questions which keep bothering us so we would like to do that in a systematic way as much as possible to begin with in one single place later on we can't do all like that as soon as a new topology comes we can keep asking this question for that topology according to the time availability or our mood and then there are many other kinds of questions also I can ask whether it will be persistent on a taking close subspaces or open subspaces instead of arbitrary subspaces so there are modifications of such questions also so this is what I mean by studying the properties of the topological property. So let us tentatively make a few definitions if we need more and more definitions or modify these definitions we can keep doing that first property is that hereditaryness so P is called a property P is called hereditary if whenever X possesses it all subspace it should possess the same property it will be cohereditary whenever X possesses it all quotients of X should possess it similarly a product invariant there are three different versions here one is finite product invariant another is countable product invariant the third one is product invariant without any quantifier qualifier okay so that is most general what is the meaning of product invariant whenever a family depending upon finite or countable family of X size are given such that if each X i has a property then the product should have and conversely the product has it each factor should also have it okay sometimes people do not bother about this one whenever X size is product has it is called product invariant so the other one they may call it as factor invariant once again there are variants of these concepts that is what I want to tell you so depending upon the author and concepts and you know what exactly you want to study and so on okay there are variants like I already told you hereditary instead of general things you can just take open subspaces you may call it open hereditary or it may be true for only close subspaces then you may call it a close hereditary and so on so they are weaker than actual being hereditary so we may call together they may call it as weekly hereditary but then just really hereditary there will be still ambiguity in this definition similarly for co-hereditary under any quotient map if the property persists then you may call it a co-hereditary but suppose it is only true for open quotients then you may call it weekly hereditary or some other people may call only closer quotients then they may call it weekly hereditary and so on okay so there are various versions of this so here is a table okay which will give you fairly good idea of whatever you are going to do right now and a little more in the in this chapter true maybe in the next section and so on so here I have listed seven of the properties that we have studied so far compactness lindelofness first and second countable separable connected path connected then these are the three properties of the property that we like to study hereditaryness over-hereditaryness and productivity between productivity there are three types i have to define had productivity countable productivity and arbitrary productivity so this being a little more complicated we will do it next time so today let us see whether we will cover this much okay so what i will do first i will take hereditaryness for all these things one after another then you go to co-hereditaryness one after another okay so let us look at compact spaces you already know that closed interval is compact but the open interval is subspace it is not compact so compactness is not hereditary on the other hand you have also proved that every closure subspace of a compact space is compact we have already proved such a thing right so that means it is weakly hereditary exactly same thing goes for lindelofness also every closed subspace of a lindelof space is lindelof right have you seen that lindelofness is not hereditary at all do you remember when we have done that can you see it easily with some example what is to be done take any lindelof space so that there is a subspace which is not lindelof what is lindelofness every open cover should have a countable sub cover space which is not lindelof we have seen that you can take for example an uncountable set and a discrete topology on that that is the easiest example of non-lindelof space okay now instead of taking subspaces you make this one in a larger space by putting one extra point namely a syrpinsky's point okay as soon as you put a syrpinsky's point what is the meaning of that that extra point the only open set containing that point is a whole space therefore when you take a covering every covering of this space must have the whole space as one of the members therefore that singleton subspace they said that will be a cover you take the text member x that's a cover so it's automatically lindelof so this is the space looks like as if we are cheating but that's a you know legal example of a lindelof space such a subspace is not lindelof you can cook up more pleasant examples or or more unpleasant examples also if you like okay so let us keep going first countable and second countable okay this is easy to check that both first and second countability are both hereditary let me do it for second countable first countable you can do exactly same way take a countable base for a topology okay now take a subspace what are the open subspace subspace take any open subset in the original thing intersect with a subspace suppose y is the subspace of x now b is a countable base for y take any member of b intersect with y now collect them that will become a countable base for y that's all okay so so up till here we have come that countability and second countability are hereditary separability connectivity path connectivity they are not hereditary connectivity and path connectivity you know already okay you take an interval remove a point it's gone the connectivity is gone right so subspaces are hardly need to be connected for a connected space and path connected space but for a separability how do you do that why separability is not hereditary remember separability is what there is a countable dense countable dense set when you go to subspace this is dense set may go away that may not mean the subspace right but that doesn't mean that there is no other countable subset so how do you how do you come how do you give an example of a space such that subspace is not separable so perhaps you may try to do similar to slindle ofness start with a non-separable thing and then hook up something bigger which is separable you know that may work no so there are ideas you have to you have to sometimes think about these things right so I have given you an example here remember we had this semi open interval topology in which the basic open subset where a comma b a closer b open so this was left semi interval so rl on the on the space of on the set of real numbers okay consider semi open interval let us see okay this is separable you can check that namely again set of rational numbers will give you a dense set okay though this is this is a this topology is larger than the visual topology so this is a dense subset so this is separable once you have dense set in x so a is a dense set a cross a will be dense inside x cross x so x cross x is separable okay but now I look for a nice subspace here namely the anti diagonal the line given by x plus y equal to 0 okay in the usual topology this is homeomorphic to r but in the in this in this separable in this semi interval topology in the product what happens this becomes a discrete space every point is open now why because given any point x comma minus x you can take x plus x to x plus epsilon and minus x to minus x plus epsilon take the product so this is half half closed rectangle sitting on the point x one of the corners will be on x comma minus x okay so this rectangle half closed rectangle will open its intersection with the the line should be open but intersection if the line is just single point so all the single points are open that means it is discrete a discrete set with a uncountable set a discrete topology and uncountable set right so that cannot be separable okay this example is quite peculiar I will I will use it again okay in the next chapter so as I have pointed out connectivity and path connectivity they are not hereditary that is seen easily okay now let us go to cohereditaryness compactness and lindelofness we have seen that the quotients are also compact and lindelof okay if you won't remember it you can just argue so take an open covering for the quotient inverse image will be an open covering for the for the original space take a finite covering and come back okay so that's it so three and four are what first countability and second countability right so these are weakly cohereditary in the sense that if it is an open coefficient it's open coefficient then x is second countable implies y is second countable once again it's very easy what you have to start with a countable base for b take f of b or b range over this curly b because f is open yeah b is open here first of all that is the crux of this now we can verify that this is a base for the topology on y so same thing works for the local base at a point also all right so open quotients are preserving the first countability and second countability so this is weakly cohereditary all right in general what happens once again we have to cook up examples here okay to examples if you do it for second countable for the whole space or even for a point the proofs will be more or less same okay so I will prove it for now this time first countability okay a space which is first countable but the quotient is not first countable quotient means what now general quotient not open quotient open quotient will be first countable okay all that I do is take infinitely many copies of r disjoint copies is it first countable any first countable space if you take any number of them and take disjoint union it will be still first countable the disjoint union does not disturb local properties all right so it is first countable now what I do I construct a quotient of this by identifying all the zeros in each copy to a single point so you can name them as r cross 0 r cross 1 r cross 2 r cross 3 and so on if you want okay these are the copies of r right now 0 cross n all of them will be identified to one single point we denote it as 0 no other identification this is all okay so equivalence classes what whenever it is 0 cross something it is equivalent to 0 cross that all other points are singleton classes so that is the quotient set give the quotient topology what is the quotient topology something is open in the quotient space if and only if its interest its inverse image is open in the disjoint union of all these copies of r okay so that is my example here x equal to disjoint union of x n countably many copies of r and why to be the quotient space obtained by identifying all the zeros to a single point the next is second countable but why is not first countable even at that point 0 which is the class of all the zeros okay the first countability fails okay so if it is not first countability cannot second countable so it this gives you example for both of them okay how to see that it is not first countable that is also easy suppose we have a countable base b n of our neighbourhoods of 0 okay local base space for this quotient space y okay let us tentatively denote this quotient space quotient map from x to y by q all right given any b n like b 1 you pick up then I can choose an interval in the x 1 x 1 copy of r around 0 so that it is so small that this b 1 intersection q of x 1 the image of that line okay that is not contained inside this q of i 1 do the same thing for all n b 1 b 2 b n and so on you look at the corresponding b n and then go to the corresponding copy of of r there you choose a neighborhood very small so that that neighborhood contain this that is all okay so once you have got for each each n this interval i n you take u to be this disjoint union of all the i n's in the disjoint union okay when you take q of those things that will be an open subset because the inverse image will be precisely disjoint union of these i n's okay so u equal to union of all these q i n's all right that is an open subset containing the point the 0 this 0 bracket you know the class of zeros but because of the choice is choice none of the b n's will be contained inside you okay if it is a base then some b n must be contained inside every open subset there must be a point in the there must be a member in the base which goes inside that so this is not a base in fact what we have proved here is no neighborhood system okay can be countable or take a countable set of neighborhood it cannot be a base that is what we have proved i started countable i never used that countable set of neighborhoods of zero okay then i showed that it is not a base all right once again five six seven what are they separability and path connect connectivity connectivity and path connectivity right so separability is easily seen to be now co-hereditary you see it was not hereditary but it is co-hereditary why take a countable set which is dense go to the quotient take the image that will be also countable set claim is that is dense very easy because all that you have to do is take an open set in the quotient space you have to show that it intersects this image right go to the inverse image that is an open subset in x so that will intersect the original dense set that point will be in the intersection of these two in the in the image that's all okay six and seven path connectivity connectivity we have seen that image just the image of a continuous function itself is has this property connected image will be connected path connected immediately path connected so it's much more stronger than being co-hereditary so the two columns we have seen accept a few things like or two examples like this we had all these things we had seen so it is like a you know this is like a summary of whatever we have done so far accept this example and this this example okay so so now we come to the the third column here product what we have seen is product of finitely many compact sets is compact have you seen that yes but lindelow space one doesn't know finitely but productivity as such okay means that it should be arbitrary product also so here I am saying yes but we don't have proof for arbitrary products not even countable products finite products you know similarly for lindelow we don't know okay the countability first countability again finitely many copies it is even you should take countabilities okay but arbitrary product we don't know okay similarly second countability the separability is more mysterious what we have done is for path connectivity we have seen this one remember connective path connectivity is easy connectivity also you have seen only for finite thing we have seen right so partly many of these things we have seen but we don't know all the full answers none of them properly right so we shall take up this one next time all right so let us stop here