2.01 Topological spaces, continuous maps


Below the video you will find accompanying notes and some pre-class questions.


(0.00) This is the first in a sequence of videos about topological spaces, aimed at people who have already seen the theory of metric spaces.

Recall the following lemma from the theory of metric spaces.

(0.49) If \(X,Y\) are metric spaces and \(F\colon X\to Y\) is a map then \(F\) is continuous if and only if \(F^{-1}(U)\) is open for all open sets \(U\subset Y\).

(2.00) The key point here is that continuity of the map \(F\) can be formulated purely in terms of open sets, with no \(\epsilon\)s or \(\delta\)s in sight. To go back to the usual definition of continuity, the way the \(\epsilon\)s and \(\delta\)s reappear is in the definition of the open sets: an open ball in a metric space is specified by its centre and its radius (the radius will be the \(\epsilon\) or \(\delta\)).

Topological spaces

(3.23) We want to turn the previous lemma into a definition (that a map is continuous if the preimages of open sets are open). For that, we need to work in a context where the notion of an ``open set'' is defined. The most general context where open sets make sense is that of a topological space.

(3.40) A topology, \(T\), on a set \(X\) is a collection of subsets of \(X\) satisfying some requirements (below). The sets in \(T\) will be called the open sets of the topology, and the requirements below are the bare minimum we need in order for these to behave in more-or-less the way we expect from the theory of metric spaces:

  • \(\emptyset\in T\), \(X\in T\),
  • (5.10) Arbitrary unions of sets in \(T\) are still in \(T\), i.e. given a collection \(\{U_i\}_{i\in I}\subset T\) where \(I\) is an indexing set then \(\bigcup_{i\in I}U_i\in T\). Arbitrary means that \(I\) can be any set (infinite, uncountable, anything). This is because unions of open sets are always open.
  • (6.42) Finite intersections of sets in \(T\) are still in \(T\), i.e. given a finite collection \(\{U_i\}_{i\in I}\subset T\) where \(I\) is a finite indexing set then \(\bigcap_{i\in I} U_i\in T\). Only finite intersections are assumed open because, for example, it is possible to find infinite collections of open sets in metric spaces (e.g. \(\mathbf{R}\)) such that the intersection fails to be open.

(8.00) A topological space is a set \(X\) equipped with a topology \(T\) on \(X\).

(8.42) The same set \(X\) can have many different topologies.

(9.05) For any set \(X\), the indiscrete topology is the smallest topology you could write down: \(T=\{\emptyset,X\}\). Certainly these two subsets need to be included in \(T\), and if you take intersections or unions of \(\emptyset\) and \(X\) then you either get \(\emptyset\) or \(X\), so it is a topology. But it is not a very useful topology: it has very few open sets (we say it is coarse).

(10.14) For any set \(X\), the discrete topology is the largest topology you could write down: \(T=\{\mbox{all subsets of }X\}\), sometimes written \(T=PX\) (the powerset of \(X\)). This has lots of open sets (we say it is fine or refined). You should think of \(X\) equipped with the discrete topology as just being a disjoint collection of points: any point is an open set.

Continuous maps

(12.00) Given topological spaces \((X,S)\) and \((Y,T)\), a map \(F\colon X\to Y\) is called continuous if \(F^{-1}(U)\in S\) for all \(U\in T\) (i.e. if \(F^{-1}(U)\) is open in \(X\) for all open sets \(U\subset Y\)).

(13.10) Given topological spaces \(X,Y,Z\) and continuous maps \(F\colon X\to Y\) and \(G\colon Y\to Z\), the composition \(G\circ F\colon X\to Z\) is continuous.

(14.43) Let \(U\subset Z\) be an open set. Its preimage under \(G\circ F\) is \((G\circ F)^{-1}(U)=F^{-1}(G^{-1}(U))\). Since \(G\) is continuous and \(U\) is open, \(G^{-1}(U)\) is open. Since \(F\) is continuous and \(G^{-1}(U)\) is open, \(F^{-1(}G^{-1}(U))\) is open. Therefore \((G\circ F)^{-1}(U)\) is open and \(G\circ F\) is continuous.

In the next video, we will see more examples of topological spaces and ways to construct new topological spaces out of old ones.

Pre-class questions

  1. Can you think of an infinite collection \(U_i\), \(i\in\mathbf{N}\), of open sets in \(\mathbf{R}\) such that \(\bigcap_{i\in\mathbf{N}}U_i\) is not open?
  2. I gave no proof for the first lemma (a map \(F\) between metric spaces is continuous if and only if \(F^{-1}(U)\) is open for every open set \(U\)) because I was assuming you have seen it before. Either by thinking for yourself or by looking at your old notes (from Analysis 4), remind yourself how to prove this.


CC-BY-SA, Jonny Evans 2017