Hausdorff spaces

Definition

Hausdorff space

We say that the space $X$ is Hausdorff if for any $x,y\in X$ such that $x\ne y$ , there are disjoint neigborhoods $U,V$ of $x,y$ , respectively.
Examples
- If $d$ is a metric on $X$ , $(X,\tau_{d})$ is Hausdorff.
- Any totally ordered set with the order topology is Hausdorff.
- The product of two Hausdorff spaces is Hausdorff.
- Any subspace of a Hausdorff space is Hausdorff.
- The Sierpiński space is not Hausdorff.
- Is the cofinite topology on $\mathbb{N}$ Hausdorff?

Points in Hausdorff

Every singleton in a Hausdorff space $X$ is a closed set.
Proof

For $x\in X$ , let $y\not\in\{x\}$ . If $U,V$ are disjoint neigborhoods of $x,y$ respectively, then $V\subseteq X-\{x\}$ . Hence $X-\{x\}$ is open.
Limit points in Hausdorff

Let $X$ be a Hausdorff space and $A\subseteq X$ . Then $x\in X$ is a limit point of $A$ if and only if every neigborhood of $x$ contains infinitely many points of $A$ .
Proof

We say that a space is a $T_{1}$ space if for any two different points $x,y$ , each has a neighborhood that does not contain the other point. Show that all singletons are closed in $X$ if and only if $X$ is $T_{1}$ .
Show that If $X$ is Hausdorff, then $X$ is $T_{1}$ . Give an example of a $T_{1}$ space that is not Hausdorff.
We say that a space is a $T_{0}$ space if for any two different points $x,y$ , one of them has a neighborhood that does not contain the other point. Show that a $T_{1}$ space is $T_{0}$ . Give an example of a $T_{0}$ space that is not $T_{1}$ . Give an example of a space that is not $T_{0}$ .