Normal Space

Definition

A Topological Space $(X,\mathcal{T})$ is normal ($T_4$) space if it satisfies T1 Axiom and for all disjoint closed sets $A,B$ in $X$, there exists disjoint open sets containing $A$ and $B$ respectively. $\forall (X-A),(X-B)\in \mathcal{T}\text{s.t.}A\cap B=\varnothing ,\exists U,V\in \mathcal{T}\text{s.t.}U\cap V=\varnothing ,A\subset U,B\subset V$

Or equivalently, $\forall A,U\text{s.t.}(X-A)\in \mathcal{T},A\in U\in \mathcal{T},\exists V\in \mathcal{T}\text{s.t.}A\subset V\text{and}\bar{V}\subset U$

Examples

A lower limit topology ${\mathcal{T}}_l$ generated by ${\mathcal{B}}_l=\{[a,b)\subset \mathbb{R}|a<b\}$ on real numbers $(\mathbb{R},{\mathcal{T}}_l)$ is normal space.

Facts

A regular Lindelof Space is a Normal Space.

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A metrizable Topological Space $(X,\mathcal{T})$ is Normal Space

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A Compact Hausdorff Space is a normal space.

A Topological Space defined by a Well-Ordered Set $X$ and an Order Topology on it is a normal space.

A closed Subspace Topology of a normal space is a normal space.

Consider a separable normal space $(X,\mathcal{T})$. If a subset $A\subset X$ of $X$ satisfies $|A|\ge \mathfrak{c}=2^{{\aleph }_0}$, then $A$ has a Limit Point in $X$.