Hausdorff Space

Definition

A Topological Space $(X,\mathcal{T})$ is Hausdorff space if $\forall x,y\in X\text{s.t.}x\ne y,\exists {\mathcal{N}}_x,{\mathcal{N}}_y\text{s.t.}{\mathcal{N}}_x\cap {\mathcal{N}}_y=\varnothing$ where ${\mathcal{N}}_x$ and ${\mathcal{N}}_y$ are the neighbourhoods of $x$ and $y$ respectively.

Facts

Every finite point set in Hausdorff space is closed ($\because$ T1 Axiom).

Consider a Topological Space $(X,\mathcal{T})$ satisfying $T_2$ axiom. Then, $X\setminus {\bigcup }_{i=1}^n{x}_i\in \mathcal{T}$ where $x_i\in X$

Consider a $T_2$ Topological Space $(X,\mathcal{T})$. Then, a Sequence of points of $X$ converges to at most one point of $X$. $\forall (x_n{)}_{i=1}^{\infty }\subset X\text{and}\forall x,y\in X,(x_n\to x\text{and}{x}_n\to y)\Rightarrow x=y$

Every totally ordered set is a Hausdorff space in the Order Topology.

A Subspace Topology of a Hausdorff space is a Hausdorff space (Hereditary Property).

The Product Space of any collection of Hausdorff spaces is a Hausdorff space.

A Compact Hausdorff Space is a normal space.

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