General Topology/Separation

Suppose we have a topological space $X$ and two distinct points $x,y\in X$. Suppose further that using the topological structure, we try to separate these points, that is, to find sets which contain one of the points, but not the other. Usually this is not possible. But the topological spaces where it is possible form important classes of topological spaces, which are then said to satisfy separation axioms. There is an important hierarchy of separation axioms which are numbered T₀ until T₆, and then there are the R₀ and R₁ axioms. The T axioms concern separation in the classical sense, whereas the R axioms concern only separation of topologically distinguishable points.

Alternatively, we may define a T₀ space to be a space where any two points are topologically distinguishable.

The situation is depicted in the following picture:

The situation is depicted in the following picture:

We introduce our first R axiom:

By far the most common class of spaces that satisfy a separation axiom are the Hausdorff spaces:

This situation is depicted in the following picture:

Template:Proposition

Hausdorff spaces are located within the T₀ until T₆ hierarchy:

There are also stronger separation conditions. They are usually formulated in regard to closed sets, and hence they don't really fit in with the T-axiom hierarchy (you'll see what I mean), but combining them with a small T axiom (namely T₀ or T₁), we produce corresponding axioms in the T-axiom hierarchy.

The situation is depicted in the following picture:

Exercises

1. Let $X$ be a finite set. Prove that there exists exactly one Hausdorff topology on $X$ and find out what it is.
2. Let $X$ be a topological space. Prove that the sets $U\subseteq X$ such that $X\setminus U$ is finite, when the empty set is adjoined to it, form a topology on $X$ (this topology is called the co-finite topology). Further prove that if $X$ is infinite, then together with this topology it is $T_1$, but not Hausdorff.
3. Let $X$ be a topological space that is equipped with the initial topology stemming from a family of functions $f_{\alpha }:X\to X_{\alpha }$. Prove that $X$ is a T₀ space if and only if for each $x\ne y$ in $X$, there exists an $\alpha$ such that $f_{\alpha }(x)$ and $f_{\alpha }(y)$ are topologically distinguishable.