Hausdorff Spaces

Definition (Hausdorff space)

A topological space $X$ is said to be a Hausdorff space if, for any two distinct points $x,y\in X$, there exist open subsets $U,V\subset X$ such that $x\in U$, $y\in V$ and $U\cap V=\varnothing$.

mention $T_2$

Every one-point set in a Hausdorff space is closed.

The converse is false. TODO: find counterexample

Every finite point set in a Hausdorff space is closed.

A topological space is a Hausdorff space iff each net converges to at most one point.

Hausdorff & subspaces Hausdorff & initial topology Hausdorff & products

compact subsets of Hausdorff spaces

Fixed Points in Hausdorff Spaces

Recall that a fixed point of a function $f:X\to X$ is a point $x\in X$ with the property that $f(x)=x$.

Theorem (Fixed Point Set is Closed)

If $f:X\to X$ is a continuous function on a Hausdorff space $X$, then the set of fixed points of $f$ is a closed subset of $X$.