Subjects: Topology
Links: Separation Axioms, Fréchet Spaces, Separation of Sets, Topological Subspaces, Hausdorff Spaces, Continuous Functions and Homeomorphims, Limit and Colimit Topology

Def: A topological space $(X,\tau )$ is normal if for any two disjoint closed subsets of $X$, $F$ and $G$ , there are disjoint open sets $U$ and $V$ such that $F\subseteq U$ and $G\subseteq V$.

Def: A topological space $(X,\tau )$ is a $T_4$ space if it is normal and a $T_1$ space.

Obs: A $T_4$ space is also $T_0,T_1,T_2,$ and $T_3$.

Prop: Let $(X,\tau )$ a $T_1$ space. The following are equivalent

• The space $X$ is $T_4$
• for any closed subset $F\subseteq X$ and $A\in \tau$ such that $F\subseteq A$, then there's $B\in \tau$ such that $F\subseteq B\subseteq \text{cl}(B)\subseteq A$.
• For every $U,V\in \tau$ such that $U\cup V=X$, then there are closed subsets $F$ and $G$ of $X$, such that $F\subseteq U$, $G\subseteq V$, and $F\cup G=X$.

Urysohn's Lemma: For every pair $A,B$ of disjoint closed subsets of a $T_4$ space $X$ there exists a function $f:X\to I$ such that $f(x)=0$ for $x\in A$ and $f(x)=1$ for $x\in B$.

Cor: A topological space $X$ a $T_4$ space iff for any two disjoint closed set $F$ and $G$, there exits a continuous function $f:X\to I$ such that $f[F]=\{0\}$ and $f[G]=\{1\}$.

Cor: A subset $A$ of a $T_4$ space $X$ is a closed $G_{\delta }$-set iff there exists a continuous function $f:X\to I$ such that $A=f^{-1}\{0\}$.

Cor: A subset $A$ of a $T_4$ space $X$ is an open $F_{\sigma }$-set iff there exists a continuous function $f:X\to I$ such that $A=f^{-1}[(0,1]]$.

Obs: Every zero set is a closed $G_{\delta }$ and a co-zero set is an open $F_{\sigma }$.

Meaning that closed $G_{\delta }$ and open $F_{\sigma }$ on a $T_4$ space are precisely the zero sets and co-zero sets.

Lemma: If $X$ is a $T_1$ space and for every closed set $F\subseteq X$ and every open $W\subseteq X$ that contains $F$ there exists a sequence $W_0,W_1,W_2,\dots$ of open subsets such that $F\subseteq \bigcup _{n<\omega }{W}_n$ and $\text{cl}(W_n)\subseteq W$ for $n<\omega$, then $X$ is a $T_4$ space.

Th: Every second countable $T_3$ space is a $T_4$ space.

Th: Every countable $T_3$ space is a $T_4$ space.

Th: Let $X$ be a topological space. $X$ is normal iff every point-finite open cover of $X$ admits a shrinking.

Th: If $X$ is a normal space and $f:X\to Y$ is a closed, continuous and surjective function, then $Y$ is normal.

Prop: If $Y$ is a normal space and $f:X\to Y$ is a closed, continuous and surjective function. the $X$ is normal.

Prop: Normality is hereditary with respect to closed sets.

Prop: Normality is hereditary with respect to $F_{\sigma }$ sets.

Tietze Extension theorem: Every continuous function from a closed subspace $M$ of $T_4$ space $X$ to $[0,1]$ or $\mathbb{R}$ is continuously extendable over $X$.

Cor: If a continuous mapping of a dense subset of a topological space $X$ to a $T_2$ space $Y$ is continuously extendable over $X$, then the extension is uniquely determined by $f$.

Cor: Let $f$ be a continuous mapping fo a closed set $F$ of a normal space $X$ into ${\mathbb{S}}^n$. Then we can continuously extend $f$ over an open cover of $X$ which contains $F$.

Cor: No separable $T_4$ space contains a closed discrete subspace of cardinality $\mathfrak{c}\mathfrak{=}{\mathfrak{2}}^{{\mathrm{\aleph }}_{\mathfrak{0}}}$.

Prop: Let $\mathcal{U}$ be a locally finite open covering of a normal space $X$. Then $\mathcal{U}$ has a $\sigma$-discrete refinement and a $\sigma$-discrete closed refinement.

Th: For every countable discrete $\{F_n:n<\omega \}$ of closed subsets of a $T_4$ space $X$ there exists a family $\{U_n:n<\omega \}$ of open subsets of $X$ such that $F_n\subseteq U_n$ for $n<\omega$ and $\text{cl}(U_n)\cap \text{cl}(U_m)=\varnothing$ for $n\ne m$.

Th: Let $X$ be a topological space. If $\{X_n\mid n<\omega \}$ is an increasing family of closed normal subspaces of $X$ such that $X=\bigcup _{n<\omega }{X}_n$, then $X$ is normal.

This result can be put in the languages of category theory theore as:

If $(i_n:X_n\to X_{n+1}{)}_{n<\omega }$ is a sequence of closed embeddings between normal spaces, then the colimit $\underrightarrow{\lim }{X}_n$ is also normal.

Hereditarily Normal

Def: A space $X$ is hereditarily normal if every subspace is normal. A space $X$ is $T_5$ if it is hereditarily normal and $T_1$.

Obs: $T_5$ is a hereditary property.

Th: For every $T_1$-space $X$ the following conditions are equivalent:

• The space $X$ is $T_5$.
• Every open subspace of $X$ is $T_4$.
• For every pair of separated sets $A,B\subseteq X$, then $A,B$ are separated by neighbourhoods.

Lemma: The space $[0,\omega {)}^{{\omega }_1}$ is not a normal space.

Cor: Let $\{(X_{\alpha },{\tau }_{\alpha })\mid \alpha <\kappa \}$ be a family of not indiscrete topological spaces. If $\kappa >\omega$, then $\prod _{\alpha <\kappa }{X}_{\alpha }$ is not hereditarily normal.

One can be tempted to compare Collectionwise Normal Spaces to hereditarily normal spaces, but these properties are completely independent of each other.

Perfect Normality

Def: A topological space $X$ is called a perfectly normal space if $X$ is a normal space and every closed subset of $X$ is a $G_{\delta }$-set.

Def: A topological space $X$ is called a $T_6$ space if $X$ is perfectly normal and $T_1$.

Obs: A second countable $T_4$ space is a $T_6$ space.

Vedenisoff Theorem: For a $T_1$ space the following conditions are equivalent:

• The space $X$ is $T_6$.
• Open subsets of $X$ are co-zero sets
• Closed subsets of $X$ are zero sets
• For a pair of disjoint closed subsets $A,B\subseteq X$, then they are precisely separated by a continuous function.

Th: The class of $T_6$ spaces are invariant under continuous closed mappings.

Prop: $T_6$ is hereditary.