2.01 Topological spaces, continuous maps

Video

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

Notes

(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.