Hausdorff Spaces

Definition: Hausdorff Space

A topological space $(X,\tau )$ is a Hausdorff space or $T_2$ space if each two distinct points $p_1,p_2\in X$ have disjoint neighborhoods.

$$\forall p_1,p_2\in X:\exists N(p_1),N(p_2):N(p_1)\cap N(p_2)=\varnothing$$

Theorem: Finite Subsets of Hausdorff Spaces are Closed

Every finite subset of a Hausdorff space is closed.

PROOF

TODO

Theorem: Open Subspaces of Hausdorff Spaces

Every topological subspace from an open subset of a Hausdorff space is itself a Hausdorff space.

PROOF

TODO

Theorem: Limit Uniqueness in Hausdorff Spaces

Let $(X,\tau )$ be a topological space and let $(x_n{)}_{n\in I}$ be a sequence of points in $X$.

If $(X,\tau )$ is Hausdorff and $(x_n{)}_{n\in I}$ is convergent, then it has only one limit:

$$\underset{n\to \infty }{\lim }{x}_n=L\text{ and }\underset{n\to \infty }{\lim }{x}_n=L^{\prime }⟹L=L^{\prime }$$

PROOF

TODO

Theorem: Infinite Neighborhoods of Accumulation Points

Let $S$ be a subset of a Hausdorff space $X$.

If $p\in X$ is a limit point of $S$, then every neighborhood of $p$ contains infinitely many points of $S$.

PROOF

TODO