 Welcome to chapter 2, points of topology course part 1. In this chapter we discuss a number of topological properties which can be classified as or which can be called smallness properties. So what are those I have listed them here, path connectivity, connectivity, compactness, lindelof property, separability, first and second countability. It is better to leave these phrase smallness properties undefined, it is not a part of mathematics ok, it is about mathematics. So there is no need to define this word rigorously, let it stand as a you know whatever it implies in the English, English words there, however as a temporary stop cap you can explain it as something ok, roughly speaking we just mean that the topology does not have too many open sets, but that is as we as the the words themselves. So I will try to make a definition, but do not worry about that definition much, just try to just to explain that to what they saw. So you can call a topological property a smallness property if whenever x tau satisfies p for all topologies tau prime on x such that this tau prime is smaller than tau and tau prime is contained as tau, x tau prime also satisfies it. So in that sense with respect to this p tau is quite small after that everything which is smaller than that will definitely satisfy bigger ones you do not know that is the meaning of this ok. So with this definition you can keep track that many of these properties here do fit into this definition, I mean they satisfy the property. I caution you the last two or not of that nature nevertheless they fit into the vague definition you will see why ok, so that is why I am not very fussy about this definition ok. So the first module here is module 32 path connectivity which is directly from layman's point of view or what may say elementary geometric concepts which are generalized into path connectivity. So first of all let us make a rigorous definition what do we mean by path start with any voluptuous space by a path sometimes called a curve or sometimes called an arc in x etc. So I will be using the word path only quite often we just mean a continuous map this gamma from a closed interval a b to x the points gamma a and gamma b are called end points of gamma to be more specific you can call gamma a the initial point and gamma b the terminal point we also say that gamma is a path joining gamma a which is z 1 and gamma b z 2 so all these terminologies which are very very much just English words they have been given some definite meaning here that is all if such a path exists we say z 1 and z 2 can be joined in x is that clear you may think that a curve should be something it must have some positive length and so on with this definition there is no notion of length and so on because x is only a topological space before you want to talk about length etc you must have a metric there secondly even if you try to draw some picture of this picture means what then you have to assume x is arc or something right the first thing is this is just a continuous map okay so in particular this may be a constant function if you look at the image it is just a single point so you may say that this is not a definition of a path at all before throwing it away like that you have to wait why we have this such a general definition as a path a point map is also treated as a path it is called a constant path it is all so let us have gone to that there is no need to throw it away okay though it is defying our common sense alright so let us carry on with these definitions given two path is now path is served from some closed interval so there are two of them means gamma i from a i b i to x okay i equal to 1 and 2 suppose that gamma 1 of b 1 is equal to gamma 2 of a 2 in other words end point of gamma 1 is the beginning point initial point of gamma 2 terminal point of the first one is equal to the initial point of gamma 2 then we define the composite path okay this composite path is not a composition of of maps composition of functions so we have to pay attention to that so I am using this dot here gamma 1 dot gamma 2 from some interval a b which I am going to define into x everything is inside x by first taking a a starting point a 1 here I go all the way to b 1 via gamma 1 okay so gamma 1 will be the map from a 1 to b 1 but now the the second point is some a 2 right so I have to shift a 2 to b 1 in the interval wherever it is and then trace the rest of the lines with gamma 2 so I take a equal to a 1 b equal to b 1 plus b 2 minus a 2 okay so b 2 minus a 2 brings a 2 or t minus a 2 whatever a 2 to 0 add b 1 so that it starts at b 1 that is so this is the shifting of the interval a 2 b 2 okay to start at b 1 after doing that a and a is and we are defined like this gamma t is the first part gamma 1 between a 1 and b 1 from b 1 to b 1 plus b 2 minus 2 this length is precisely equal to b 2 minus a 2 it is t plus a 2 minus b 1 then t equal to b 1 what is this will be this will be a 2 right t could be b 1 here b 1 here so this is gamma 2 of a 2 which is equal to gamma 1 of b 1 where t equal to b 1 is to gamma 1 of b 1 and this is gamma 2 of a 2 they are equal therefore this is a well-defined continuous function on a to b okay by the inverse path I am going to define what is the mean of inverse path okay of a given path we just mean the path defined by a plus b minus t we just want to reverse t to minus t so that you are just tracing the same path in the opposite direction but since you are not working into 0 comma something a comma b you have to do this much of you know arithmetic here a plus b minus t is the correct thing so when t equal to b okay it is a so that is the end point now gamma a becomes the end point when t equal to a it is gamma b which is the which is the terminal point which is starting point so the initial point and the and the and the terminal points are interchanged so that is the path the inverse path of of this one traverse in the opposite direction these definitions are borrowed from what happens to when you do integration on paths unfortunately on an arbitrary path is you can't do an integration you have to have differentiable paths or piecewise differentiable paths but we don't need that in topology or first of all we don't know what is the meaning of differentiability function taking values in an arbitrary topographic space that will be too much we we are doing topology here even in a metric space you won't know what to do with that okay so we have come far away from the Euclideanness of these topological spaces wherein differentiability etc are also valid but the idea is borrowed from there often by a curve one means the image of gamma people always you know think of arc of a circle okay ellipse or a parabola see in each of the cases the whatever curve you mean is described in a different way the circle or arc of a circle the entire circle is described by quadratic equation parabola is described quadratic equation we want to get rid of all that and that forces us to have this parameterization okay so gamma which is described in this curve now is called a parameterization like e going to t going to e power i t two power i t whatever is the description or parameterization of the circle okay so this is called a parameterization by those people who know what is a curve by different definition okay for instance the circle third equal to mod z equal to 1 this is another way of describing the unit circle this can be thought of as a curve given by gamma theta to cos theta sin theta or equivalent to e power two by i theta number i just e power i theta okay so that is a parameterization of the circle however we will consider two paths should be different if they are given by different functions okay the only condition of the function that we put is that it must be continuous so such things are called paths observe that since gamma two of b1 plus a2 minus b1 is gamma two of a2 gamma three by one it follows that the composites well defined which I have already told you thus we see that composite path is obtained by first traversing along gamma one and then along gamma two so this is the geometric idea behind this definition moreover image of this gamma is nothing but image of gamma one union image of gamma two they may overlap they may whatever may have both of them may be constant at some point then if that is the case that point will be the same for both of them because one point they agree here so all those things are allowed very very very generic nature of the definition here also note that if gamma from a b to x is a path and a less than a could a1 less than a could a2 less than a3 to b suppose we take a division cut the interval into two parts by taking a point between that namely a2 okay then you can think of this gamma as gamma one dot gamma two where gamma i's are the restriction of the original gamma to the sub integral a1 to a2 and a2 to a3 so this remark will be crucial practically important as far as this can be done for any only one division of apple you can do it finitely many cutoffs also finitely many division source okay so suppose you have a map from alpha prime beta prime to a b strictly increasing continuous function so that itself is a path actually because it's a continuous function it's a path where inside this space a b but it is strictly increasing and i'm assuming that alpha prime goes to a beta prime goes to b okay then we say that the curve gamma composite tau see gamma is from a b to x okay tau is from alpha prime beta prime to a b or just you can write alpha beta there is no need to prime the composite curve arises by a change of parameterization from gamma this is the definition now so earlier we had gamma as parameterization now gamma composite tau is a parameterization okay from gamma or that gamma composite tau is a reparameterization of gamma what is the condition for reparameterization this reparameterizing factor must be strictly increasing continuous function in in particular the direction which which you are tracing the curve is not changed for example the inverse part of inverse is not reparameterization okay or sorry gamma inverse which you have defined here okay gamma 2 gamma inverse we have defined that is not a reparameterization it is tracing it in the opposite direction okay so here i what i'm saying is this tau is from here to here but tau inverse from here to here any strictly monotonically increasing function which is continuous is systematically inverse is also continuous okay tau inverse is defined and is continuous follows that the change of parameterization is an equivalence relation among the set of all parties okay so this is one of the reasons why the equivalence classes are considered as as parties or curves okay not exactly you are not exactly interested in the equivalence ratio you can pick up one of the parameterizations which you which suits you so this will bring you more to the geometry of the curve rather than the parameterization itself okay but when you define such a thing you must see why you are using the parameterization and what is the intrinsic property why it won't change when you take the take this change of parameterization all this you have to be worried about okay so here are examples the mapping zeta t which is cos 2 pi i t plus i sin 2 pi i t which i was telling you can write it as exponential t or whatever 0 less than t less than 1 is another parameterization of the unit circle which was earlier theta only to cos theta plus i sin theta but the interval was 0 to 2 pi okay so from year to year you take multiplication by 2 pi and follow it then you get this one so that is all idea even any two points z1 and z2 belong into c the mapping 1 minus t times z1 plus t times z2 this is called what if t is restricted between 0 and 1 it is the line segment from z1 to z2 okay so this will be denoted by z1 to z2 this this bracket z1, z2 will be called line segment this notation is borrowed from what we do it real line but it's not necessarily line it could be in zr2 also line segment may be like this like this or something okay so this is the notation for actually I am not only denoting this one I am using this one for the path that describes this arc it must be from z1 to z2 t going to 1 minus t times z1 the domain must be 0, 1 and all that is there so you are welcome to write down various reparameterization of the same path okay so for example t going to t z1 plus 1 minus t z2 again 0 t less than equal to 1 will be the inverse because now when t equal to 0 it will be z2 t equal to 1 it will be z1 so this is the inverse path of that okay t is replaced by 1 minus t note that such notation is somewhat unusual okay other than interval scenario okay interval scenario like that this you have to be now familiar with often we shall merely refer to either of these segments merely by the line segment between z1 and z2 this will make sense only if we are working inside a vector space okay here I am talking about c so why I have done guess for c because these things are very very useful in complex analysis when you are doing component integration that is why motivation for just quoting these things that is all after all whatever psychology you study you would like to use them elsewhere also one of the most intuitive and primitive and important topological property is path connectedness okay deep time we just say no connectivity just means that there is no road there is no vehicle for here to there we are all isolated and so on it keeps saying so path there what do you mean by we have to have a path so that we can drive a vehicle we can try take a bicycle or whatever okay there must be a path we say a subset a of a topological space is path connected if any two point at once or two inside a can be joined in a remember what the meaning of joint we can find a path within a with end points as z1 and z2 okay by very definition if they can be joined means it is symmetric okay I do not care whether z1 is first one z2 is a terminal or the other way around it is a symmetric relation z1 can always be joined to z2 by the constant path so it is reflexive right so finally if you can z1 to z2 and z2 to z3 all of them inside a what happens you can take the composition that we have defined so that will take you from z1 to z3 therefore this this if you think of this as a relation that z1 can be joined to z2 that is a equivalence relation right now I have just defined what do you mean by path connectivity that just means that is one only one equivalence class z1 can be every point can be joined to another point okay if we have one single point or not inside a to which every other point of a can be joined then any two points of they can also be joined with each other what you have to do first start with z1 to z0 and then z0 to z2 so you take two of the paths like this and use the composition okay one single point which we can be joined all that okay it is like a so you get union of x-axis and y-axis to 0 0 comma 0 can be joined to every path every point in the union of x-axis and y-axis right so that is the kind of situation I am in mind here okay so now just all the time we have examples inside r2 r2 and so on the same thing can be done in any vector space v okay we call that a subset a of a vector space v is called a convex subset whenever you and we are there the entire line segment must be there right so that is the original convexity automatically what does it mean any two points can be joined by the line segment itself therefore in particular every convex set is path connected and more generally what you can take is a is called a star shaped subset if there exist a u belonging to a such set every v inside a a line segment is inside a so this is the case wherein you are taking union of say two lines which are intersecting at a point or several lines which are intersecting at a single point all of them so those things are stars shape okay they are not convex okay yet they are path connected because of this property because of this property that we have discussed here if we delete a single point from r it becomes disconnected well we haven't proved what is the meaning of disconnected we haven't even defined it but we immediately understand what is this one this is just this is just english word okay so right now let it hang it like that then we will we will define what is the meaning of disconnected then and we shall actually prove rigorously that this happens indeed this is true of any interval also if if we delete a point as a point is deleted okay that point should not be the end point if you from close interval 0 and if you delete 1 it will be still connected right so here I am meaning path connected okay because every point in the interval close interval 0 to open interval 1 say say a comma b every any two points you can join them there is a line segment is already there it is already convex right so you can use the word instead of just disconnected connected and so on path connected which we have defined then everything is clear here all right but if you delete one point from r why it is not path connected that is not obvious you have to use something deeper about real number okay we will come to that one so on try to prove that even if you delete a finite number of points from r n and greater than your pro quo it remains path connected so this is immediate this this can be done immediately okay see if you remove one point from r it gets disconnected but if you remove finitely many points any number of finitely many points from r2 it is still it is it is still connected you can join them by path how do you do that so I would like you to leave it to you as a exercise if which is if it is too difficult or you have not understood you can contact us again okay similarly the unit sphere in r n n greater than equal to 2 n equal to 2 it is the circle n equal to 3 it is the two sphere and so on they are all path connected even though they are not star shape how do you show the circle is path connected give an end to point there are two ways of you know you can have two different arcs you can use the restricted parameterization done but when you go to the two sphere how do you do that think about these things these are completely geometric and it is like an extension of your calculus study of calculus okay so at this stage we will take a break so we will continue tomorrow next time thank you