Hausdorff Spaces

Definition: Hausdorff Space

A topological space $(X,\tau )$ is a Hausdorff space iff for all distinct points $p_1,p_2\in X$ there are disjoint neighbourhoods $N(p_1)$ of $p_1$ and $N(p_2)$ of $p_2$.

NOTE

Hausdorff spaces are also known as $T_2$ spaces.

Properties

Theorem: Closedness of Compact Hausdorff Subspaces

Every subspace of a Hausdorff space $(H,{\tau }_H)$ is closed in $(H,{\tau }_H)$.

PROOF

TODO

Theorem: Finite Subsets of Hausdorff Spaces are Closed

Every finite subset of a Hausdorff space is closed.

PROOF

Let $(X,\tau )$ be a Hausdorff space.

Let $\{p_0\}$ contain only the point $p_0\in X$. By definition, given any other point $p\in X$, there exists disjoint neighbourhoods $N(p_0)$ and $N(p)$.

TODO

Theorem: Limits of Sequences in Hausdorff Spaces

If a sequence $(x_n{)}_{n\in \mathbb{N}}$ of points in a Hausdorff space is convergent, then it has a unique limit.

PROOF

TODO

THEOREM

Every subspace of a Hausdorff space is itself a Hausdorff space.

PROOF

TODO