Section 12: Topological Spaces

A topology $\mathcal{T}$ on a set $X$ is a collection of subsets of $X$ such that $\emptyset$ , $X$ , the union of any subcollection and the intersection of any finite subcollection are in $\mathcal{T}$ . A topological space is an ordered pair $(X,\mathcal{T})$ , i.e. a set $X$ and a topology $\mathcal{T}$ on $X$ .

An open set of a topological space $(X,\mathcal{T})$ is any set in $\mathcal{T}$ . We call an open set containing a given point an (open) neighborhood of the point.

A topology $\mathcal{T}'$ is (strictly) finer or larger than $\mathcal{T}$ if the former (properly) contains the latter. In this case $\mathcal{T}$ is said to be (strictly) coarser or smaller than $\mathcal{T}'$ . The terms “weaker” or “stronger” are used sometimes, but both can mean either finer or coarser depending on the context. Two topologies are comparable if one contains the other.

Examples of topologies

The (indiscrete) trivial topology on $X$ : $\{\emptyset,X\}$ .
The discrete topology on $X$ : $\mathcal{P}(X)$ .
The finite complement topology on $X$ is the collection of the subsets of $X$ such that their complement in $X$ is finite or $X$ .
The countable complement topology on $X$ is the collection of the subsets of $X$ such that their complement in $X$ is countable or $X$ .