Chapter 1: Some Basics in Topology Today we will introduce some basic concepts in topology. In this lecture, we will still stay in the continuous domain. After that, we will change gears to discrete setting, handling discrete objects (especially simplicial complexes) that we are more familiar with, and also that are more computationally friendly. References for the materials covered in this lecture include [2] (for Section 1 and 2) and [1] (for Section 3 and 4). 1 Topological Spaces Definition 1.1 (Topological space) A topological space is a set X endowed with a topological structure (a topology) T such that the following conditions are satisfied: 1. Both the empty set and X are elements of T . 2. Any union of arbitrarily many elements of T is an element of T . 3. Any intersection of finitely many elements of T is an element of T . In other words, it is a set equipped with set of subsets. In particular, each set A in T is said to be open. Its complement X n A is said to be closed. That is, a set A is closed if its complement is open. A set could be neither open nor closed (think of [0; 1) on the real line IR equipped with the standard topology that we will describe shortly). A set could also be both open and closed (think of the set X). The closure of a set A is the minimal closed set from X containing A. Given the same space (set) X, one can defined different set system T , which will then in turn lead to different topology. However, very often, in spaces (say Euclidean space, or more general metric space, or the simplicial complex we will use heavily later) we encounter, we use the standard topology equipped with the space. Hence the reference to the set system T is often omitted. This is a rather general and abstract definition. Now this looks a bit too alien and very abstract. Let us look some examples. Example 1: Simple discrete topology on n elemens. Let X be a set of n elements X = X 1 fx1; : : : ; xng, and let T = 2 . < X; T > forms a topology. This is a discrete topology . It is a simple topology. We can also consider the trivial topology on X, which is simply T = f;; Xg. Example 2: Metric topological space. Given a metric space (X; dX ), there is a natural way to put a topology on it. Let us now try to rephrase everything in the metric space. Definition 1.2 (Metric space) A metric space (X; dX ) is a set X equipped with a function d : X × X ! IR such that (1) dX (x; y) ≥ 0 for all x; y 2 X; (2) dX (x:y) = 0 if and only if x = y; 1A topology < X; T > is discrete if T = 2X. 1 (3) dX (x; y) = d(y; x) for all x; y 2 X; (4) dX (x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X. The function dX is often referred to as the distance metric on X. One of the most familiar metric d d space is the Euclidean space (X = IR ; dX ) with dX being the standard Euclidean distance in IR ; d i.e, dX (x; y) = kx − yk2. We will first define the metric topology on Euclidean space (IR ; d). For any x 2 IRd and " > 0, let B(x; ") := fy 2 IRd j d(x; y) < "g be the open Euclidean ball around x, and let B = fB(x; ") j x 2 IRd; " > 0g. Then set T to be the collection of (potentially infinite) union of any subset of open balls from B. It can be verified that the T as constructed this way indeed forms a topology on IRd. We call (IRd; T ) the standard topology on the Euclidean space IRd. Later in the class, if we refer to a topological space without describing its topology T explicitly, the we usually refer to standard topology (or a topology induced from certain standard topology) on the space. Let us now consider this topology on the real line. The topology T in this case consists of all 1 1 open intervals on the line. Note that for the real line, the intersection of the intervals (− k ; + k ), for all integers k ≥ 1, is the point 0. This is not an open set. This illustrates the need for the restriction to finite intersections. In general, A ⊂ IRd is open if for any point x inside, we can move it in arbitrary direction while still staying inside A. This definition of open balls and open sets can be extended to any metric space (X; d) by replacing the distance kx−yk with the metric distance d(x; y), which will give us a natural standard topology for any metric space. The natural topology defined on a metric space perhaps is perhaps the most important and most common topological space. We remark that in the previous example of generating the metric topology for a metric space (X; d), the collection of all open balls (with positive radius) B = fB(x; ") j x 2 IRd; " > 0g in fact forms a basis for the resulting topology (X; T ). Definition 1.3 (Basis of topology) Given a set X, a basis for a topology on X is a collection C of subsets of X (called basis elements) such that • for each x 2 X, there is at least one basis element C 2 C containing x; and • if x 2 C1 \ C2 for two basis elements C1;C2 2 C, then there exists a basis element C3 containing x such that C3 ⊂ C1 \ C2. The topology TC generated by C is the collection of unions of elements of C. It can be shown that given a basis, TC indeed is a valid topology on X. In our previous example, one can show that B satisfies the conditions of being a basis for IRd, and thus is a basis generating the topology T on IRd. Subspace topology. Finally, suppose that we have a topological space < X; T >. Given a subset Y ⊆ X, it has a natural topology on it which is inherited from T , denoted by TY defined as follows: open sets in TY is the intersection between open sets in T and Y. This is called the subspace topology on Y induced from < X; T >. For example, when we talk about topology for a surface S (or any compact set) embedded in IR3, we in fact mean the subspace topology on S induced from IR3. From now on, I will often omit the explicit reference of T and simply talk about a topological space X when the choice of T is clear. (In fact, we will mostly talk about the topology induced from a Euclidean space in this class.) 2 Remark. The topology (as well as the induced topology) in Euclidean space is the most common topological space one will encounter. The definition of topological space as sets of subsets may seem un-natural at first. In particular, the definitions of open and closed sets may be non-intuitive. Recall that we have said earlier that topology is about connectivity and about how the input space is put together from its subsets. Intuitively, this is captured by the open set and their union and intersections. Indeed, we will also use the term neighborhood of x 2 X to refer to an open set containing x. Neighborhoods intersect and indicate intuitively how they are patch together to form the topological space X. Closure, interior, boundaries etc. Given a set A ⊂ X, there are different way to define its closure and interior, e.g, using the concept of limit point. We will define it purely based on the general concepts we already know, i.e, open and closed sets of a topological space. Definition 1.4 Given a subset A of a topological space X (which could be neither open or closed), the closure of A, denoted by A, is defined as the intersection of all closed sets containing A. The interior of A, denoted by intA, is the union of all open sets contained in A. The boundary of A ⊆ X is defined as bdA = A \ X − A. It is easy to see that intA is open itself, and the closure A is a closed set itself. In general, intA \ bdA = ;, and A = intA [ bdA. If A is open itself, then bdA = A − A. Example: in IR2, an open or closed disk has the same closure, bundary, and interior. Now suppose we want to compare two topological spaces. We need a map between them, and we need a language to say that the two spaces are connected in the same way using this map. The language for this purpose is continuous function, which is one of the most important concepts in not just topology, but mathematics. We introduce it next. 2 Maps, homeomorphism, and homotopy There are different definitions of a continuous function. We give two equivalent ones below defined in the very general setting. Definition 2.1 (Continuous function) A neighborhood of a point x 2 X is simply an open set of X containing x. A function f : X ! Y is continuous at x 2 X if for any neighborhood V of f(x), there is a neighborhood U of x such that f(U) ⊆ V . See Figure 1 for an illustration. Function f is continuous if it is continuous at all points in X. Equivalently, a function f : X ! Y is continuous if for any open set V in Y, its preimage f −1(V ) is also open.