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:

\[ \lim_{n \to \infty} x_n = L \qquad \text{ and } \qquad \lim_{n \to \infty} x_n = L' \implies L = L' \]

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