Regular Hausdorff spaces

Subjects: Topology
Links: Hausdorff Spaces, Fréchet Spaces, Kolmogorov Spaces, Topological Spaces

Def: Let $X$ be a topological space we say that $X$ is regular space, if given any closed set $F$ and any point $x\in X\setminus F$, there exists neighbourhood $U$ of $x$ and a neighbourhood $V$ of $F$ that are disjoint.

Def: Let $X$ be a topological space we say that $X$ is $T_3$ space if $X$ is $T_1$ space and a regular space.

Obs: Every $T_3$ space is $T_2$, $T_1$, and $T_0$.

Prop: Let $(X,\tau )$ be a topological space.

• $X$ is regular.
• For any point $x\in X$ and any open $U$ such that $x\in U$, there's a $V\in \tau$ such that $x\in V\subseteq \text{cl}(V)\subseteq U$.
• For each $x\in X$ has a local base of neighbourhoods formed by closed subsets.
• For every closed set $F\subseteq X$, there's a family $\mathcal{B}\subseteq \tau$ such that $F\subseteq U$ for each $U\in \mathcal{B}$ and $F=\bigcap \text{cl}(\mathcal{B})$

Prop: Both regularity and $T_3$ are topological properties, meaning that are invariant under homeomorphisms.

Prop: Let $X$ be a regular space. If $\mathcal{U}$ is an open cover of $X$, then there exists an open cover $\mathcal{W}$ such that $\bar{\mathcal{W}}$ is refinement of $\mathcal{U}$.

Prop: $T_3$ and regularity are hereditary properties.

Th: If $\{(X_{\alpha },{\tau }_{\alpha })\mid \alpha <\kappa \}$ is a family of nonempty $T_3$ spaces, then the product $\prod _{\alpha <\kappa }{X}_{\alpha }$ is a $T_3$ space iff for each $\alpha <\kappa$, $X_{\alpha }$ is a $T_3$ space.

Prop: If $\{(X_{\alpha },{\tau }_{\alpha })\mid \alpha <\kappa \}$ is a family of nonempty $T_3$ spaces, then the sum $\underset{\alpha <\kappa }{⨁}{X}_{\alpha }$ is a $T_3$ space iff for each $\alpha <\kappa$, $X_{\alpha }$ is a $T_3$ space.

Prop: Let $X$ be a regular $T_0$ space, then it is a $T_2$ space.