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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{normal space} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{normal_spaces}{}\section*{{Normal spaces}}\label{normal_spaces} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak \noindent\hyperlink{TietzeEtensionAndLiftingProperty}{Tietze extension and lifting property}\dotfill \pageref*{TietzeEtensionAndLiftingProperty} \linebreak \noindent\hyperlink{CategoryOfNormalSpaces}{The category of normal spaces}\dotfill \pageref*{CategoryOfNormalSpaces} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A normal space is a space (typically a [[topological space]]) which satisfies one of the stronger [[separation axioms]]. [[!include main separation axioms -- table]] \hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions} A [[topological space]] $X$ is \textbf{normal} if it satisfies: \begin{itemize}% \item $T_4$: for every two [[closed subspace|closed]] [[disjoint subsets]] $A,B \subset X$ there are (optionally [[open subspace|open]]) [[neighborhoods]] $U \supset A$, $V \supset B$ such that their [[intersection]] $U \cap V$ is [[empty set|empty]]. \end{itemize} By [[Urysohn's lemma]] this is equivalent to the condition \begin{itemize}% \item for every two [[closed subspace|closed]] [[disjoint subsets]] $A,B \subset X$ there exists an [[Urysohn function]] that separates them. \end{itemize} Often one adds the requirement \begin{itemize}% \item $T_1$: every [[point]] in $X$ is closed. \end{itemize} (Unlike with [[regular spaces]], $T_0$ is not sufficient here.) One may also see terminology where a \textbf{normal space} is any space that satisfies $T_4$, while a \textbf{$T_4$-space} must satisfy both $T_4$ and $T_1$. This has the benefit that a $T_4$-space is always also a $T_3$-space while still having a term available for the weaker notion. On the other hand, the reverse might make more sense, since you would expect any space that satisfies $T_4$ to be a $T_4$-space; this convention is also seen. If instead of $T_1$, one requires \begin{itemize}% \item $R_0$: if $x$ is in the [[topological closure|closure]] of $\{y\}$, then $y$ is in the [[topological closure|closure]] of $\{x\}$, \end{itemize} then the result may be called an \textbf{$R_3$-space}. Any space that satisfies both $T_4$ and $T_1$ must be [[Hausdorff space|Hausdorff]], and every Hausdorff space satisfies $T_1$, so one may call such a space a \textbf{normal Hausdorff space}; this terminology should be clear to any reader. Any space that satisfies both $T_4$ and $R_0$ must be [[regular space|regular]] (in the weaker sense of that term), and every regular space satisfies $R_0$, so one may call such a space a \textbf{normal regular space}; however, those who interpret `normal' to include $T_1$ usually also interpret `regular' to include $T_1$, so this term can be ambiguous. Every normal Hausdorff space is an [[Urysohn space]], a fortiori regular and a fortiori Hausdorff. It can be useful to rephrase $T_4$ in terms of only open sets instead of also closed ones: \begin{itemize}% \item $T_4$: if $G,H \subset X$ are [[open subspace|open]] and $G \cup H = X$, then there exist open sets $U,V$ such that $U \cup G$ and $V \cup H$ are still $X$ but $U \cap V$ is empty. \end{itemize} This definition is suitable for generalisation to [[locales]] and also for use in [[constructive mathematics]] (where it is not equivalent to the usual one). To spell out the localic case, a \textbf{normal locale} is a [[frame]] $L$ such that \begin{itemize}% \item $T_4$: if $G,H \in L$ are opens and $G \vee H = \top$, then there exist opens $U,V$ such that $U \vee G$ and $V \vee H$ are still $\top$ but $U \wedge V = \bot$. \end{itemize} \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} \begin{example} \label{}\hypertarget{}{} Let $(X,d)$ be a [[metric space]] regarded as a [[topological space]] via its [[metric topology]]. Then this is a normal Hausdorff space. \end{example} \begin{proof} We need to show that given two [[disjoint subsets|disjoint]] [[closed subsets]] $C_1, C_2 \subset X$, there exist [[disjoint subset|disjoint]] [[open neighbourhoods]] $U_{C_1} \supset C_1$ and $U_{C_2} \supset C_2$. Consider the function \begin{displaymath} d(S,-) \colon X \to \mathbb{R} \end{displaymath} which computes [[distances]] from a subset $S \subset X$, by forming the [[infimum]] of the distances to all its points: \begin{displaymath} d(S,x) \coloneqq inf\left\{ d(s,x) \vert s \in S \right\} \,. \end{displaymath} If $S$ is closed and $x \notin S$, then $d(S, x) \gt 0$. Then the [[unions]] of [[open balls]] \begin{displaymath} U_{C_1} \coloneqq \underset{x_1 \in C_1}{\bigcup} B^\circ_{x_1}( \frac1{2}d(C_2,x_1) ) \end{displaymath} and \begin{displaymath} U_{C_2} \coloneqq \underset{x_2 \in C_2}{\bigcup} B^\circ_{x_2}( \frac1{2}d(C_1,x_2) ) \,. \end{displaymath} have the required properties. For if there exist $x_1 \in C_1, x_2 \in C_2$ and $y \in B^\circ_{x_1}( \frac1{2}d(C_2,x_1) ) \cap B^\circ_{x_2}( \frac1{2}d(C_1,x_2) )$, then \begin{displaymath} d(x_1, x_2) \leq d(x_1, y) + d(y, x_2) \lt \frac1{2} (d(C_2, x_1) + d(C_1, x_2)) \leq \max\{d(C_2, x_1), d(C_1, x_2)\} \end{displaymath} and if $d(C_1, x_2) \leq d(C_2, x_1)$ say, then $d(x_2, x_1) = d(x_1, x_2) \lt d(C_2, x_1)$, contradicting the definition of $d(C_2, x_1)$. \end{proof} \begin{prop} \label{}\hypertarget{}{} Every [[regular space|regular]] [[second countable space]] is normal. \end{prop} See [[Urysohn metrization theorem]] for details. \begin{prop} \label{}\hypertarget{}{} \textbf{(Dieudonn\'e{}`s theorem)} Every [[paracompact Hausdorff space]], in particular every [[compact Hausdorff space]], is normal. \end{prop} See \emph{[[paracompact Hausdorff spaces are normal]]} for details. \begin{example} \label{}\hypertarget{}{} The [[real numbers]] equipped with their [[K-topology]] $\mathbb{R}_K$ are a [[Hausdorff topological space]] which is not a [[regular Hausdorff space]] (hence in particular not a normal Hausdorff space). \end{example} \begin{proof} By construction the K-topology is [[finer topology|finer]] than the usual [[Euclidean space|euclidean]] [[metric topology]]. Since the latter is Hausdorff, so is $\mathbb{R}_K$. It remains to see that $\mathbb{R}_K$ contains a point and a [[disjoint subset|disjoint]] closed subset such that they do not have disjoint [[open neighbourhoods]]. But this is the case essentially by construction: Observe that \begin{displaymath} \mathbb{R} \backslash K \;=\; (-\infty,-1/2) \cup \left( (-1,1) \backslash K \right) \cup (1/2, \infty) \end{displaymath} is an open subset in $\mathbb{R}_K$, whence \begin{displaymath} K = \mathbb{R} \backslash ( \mathbb{R} \backslash K ) \end{displaymath} is a [[closed subset]] of $\mathbb{R}_K$. But every [[open neighbourhood]] of $\{0\}$ contains at least $(-\epsilon, \epsilon) \backslash K$ for some positive real number $\epsilon$. There exists then $n \in \mathbb{N}_{\geq 0}$ with $1/n \lt \epsilon$ and $1/n \in K$. An open neighbourhood of $K$ needs to contain an open interval around $1/n$, and hence will have non-trivial intersection with $(-\epsilon, \epsilon)$. Therefore $\{0\}$ and $K$ may not be separated by disjoint open neighbourhoods, and so $\mathbb{R}_K$ is not normal. \end{proof} \begin{example} \label{Morse}\hypertarget{Morse}{} If $\omega_1$ is the first un-[[countable ordinal]] with the [[order topology]], and $\widebar{\omega_1}$ its [[one-point compactification]], then $X = \omega_1 \times \widebar{\omega_1}$ with the [[product topology]] is not normal. Indeed, let $\infty \in \widebar{\omega_1}$ be the unique point in the [[complement]] of $\omega_1 \hookrightarrow \widebar{\omega_1}$; then it may be shown that every open set $U$ in $X$ that includes the closed set $A = \{(x, x): x \neq \infty\}$ in $X$ must somewhere intersect the closed set $\omega_1 \times \{\infty\}$ which is disjoint from $A$. For if that were false, then we could define an increasing sequence $x_n \in \omega_1$ by recursion, letting $x_0 = 0$ and letting $x_{n+1} \in \omega_1$ be the least element that is greater than $x_n$ and such that $(x_n, x_{n+1}) \notin U$. Then, letting $b \in \omega_1$ be the [[supremum]] of this increasing sequence, the sequence $(x_n, x_{n+1})$ converges to $(b, b)$, and yet the neighborhood $U$ of $(b, b)$ contains none of the points of this sequence, which is a contradiction. \end{example} This example also shows that general subspaces of normal spaces need not be normal, since $\omega_1 \times \widebar{\omega_1}$ is an open subspace of the compact Hausdorff space $\widebar{\omega_1} \times \widebar{\omega_1}$, which is itself normal. \begin{example} \label{}\hypertarget{}{} An uncountable product of infinite discrete spaces $X$ is not normal. More generally, a product of $T_1$ spaces $X_i$ uncountably many of which are not [[limit point compact space|limit point compact]] is not normal. Indeed, by a simple application of Remark \ref{retracts} below and the fact that closed subspaces of normal Hausdorff spaces are normal Hausdorff, it suffices to see that the archetypal example $\mathbb{N}^{\omega_1}$ is not normal. For a readable and not overly long account of this result, see \href{https://dantopology.wordpress.com/2009/10/13/the-uncountable-product-of-the-countable-discrete-space-is-not-normal/}{Dan Ma's blog}. \end{example} \begin{example} \label{}\hypertarget{}{} A [[Dowker space]] is an example of a normal space which is not [[countably paracompact topological space|countably paracompact]]. \end{example} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties} \begin{prop} \label{T4InTermsOfTopologicalClosures}\hypertarget{T4InTermsOfTopologicalClosures}{} \textbf{(normality in terms of topological closures)} A [[topological space]] $(X,\tau)$ is normal Hausdorff, precisely if all points are closed and for all [[closed subsets]] $C \subset X$ with [[open neighbourhood]] $U \supset C$ there exists a smaller open neighbourhood $V \supset C$ whose [[topological closure]] $Cl(V)$ is still contained in $U$: \begin{displaymath} C \subset V \subset Cl(V) \subset U \,. \end{displaymath} \end{prop} \begin{proof} In one direction, assume that $(X,\tau)$ is normal, and consider \begin{displaymath} C \subset U \,. \end{displaymath} It follows that the [[complement]] of the open subset $U$ is closed and disjoint from $C$: \begin{displaymath} C \cap X \setminus U = \emptyset \,. \end{displaymath} Therefore by assumption of normality of $(X,\tau)$, there exist open neighbourhoods with \begin{displaymath} V \supset C \,, \phantom{AA} W \supset X \setminus U \phantom{AA} \text{with} \phantom{AA} V \cap W = \emptyset \,. \end{displaymath} But this means that \begin{displaymath} V \subset X \setminus W \end{displaymath} and since the [[complement]] $X \setminus W$ of the open set $W$ is closed, it still contains the closure of $V$, so that we have \begin{displaymath} C \subset V \subset Cl(V) \subset X \setminus W \subset U \end{displaymath} as required. In the other direction, assume that for every open neighbourhood $U \supset C$ of a closed subset $C$ there exists a smaller open neighbourhood $V$ with \begin{displaymath} C \subset V \subset Cl(V) \subset U \,. \end{displaymath} Consider disjoint closed subsets \begin{displaymath} C_1, C_2 \subset X \,, \phantom{AAA} C_1 \cap C_2 = \emptyset \,. \end{displaymath} We need to produce disjoint open neighbourhoods for them. From their disjointness it follows that \begin{displaymath} X \setminus C_2 \supset C_1 \end{displaymath} is an open neighbourhood. Hence by assumption there is an open neighbourhood $V$ with \begin{displaymath} C_1 \subset V \subset Cl(V) \subset X \setminus C_2 \,. \end{displaymath} Thus \begin{displaymath} V \supset C_1 \,, \phantom{AAAA} X \setminus Cl(V) \supset C_2 \end{displaymath} are two disjoint open neighbourhoods, as required. \end{proof} \hypertarget{TietzeEtensionAndLiftingProperty}{}\subsubsection*{{Tietze extension and lifting property}}\label{TietzeEtensionAndLiftingProperty} The [[Tietze extension theorem]] applies to normal spaces. In fact the Tietze extension theorem can serve as a basis of a [[category theory|category theoretic]] characterization of normal spaces: a (Hausdorff) space $X$ is normal if and only if every function $f \colon A \to \mathbb{R}$ from a [[closed subspace]] $A \subset X$ admits an [[extension]] $\tilde{f}: X \to \mathbb{R}$, or what is the same, every [[regular monomorphism]] into $X$ in $Haus$ has the [[left lifting property]] with respect to the map $\mathbb{R} \to 1$. (See \emph{[[separation axioms in terms of lifting properties]]} (\hyperlink{Gavrilovich14}{Gavrilovich 14}) for further categorical characterizations of various topological properties in terms of lifting problems.) \hypertarget{CategoryOfNormalSpaces}{}\subsubsection*{{The category of normal spaces}}\label{CategoryOfNormalSpaces} Although normal (Hausdorff) spaces are ``[[nice topological spaces]]'' (being for example [[Tychonoff spaces]], by [[Urysohn's lemma]]), the [[category]] of normal topological spaces with [[continuous maps]] between them seems not to be very well-behaved. (Cf. the rule of thumb expressed in [[dichotomy between nice objects and nice categories]].) It admits [[equalizers]] of pairs of maps $f, g: X \rightrightarrows Y$ (computed as in $Top$ or $Haus$; one uses the easy fact that closed subspaces of normal spaces are normal). However it curiously does not have [[products]] -- or at least it is not closed under products in $Top$, as shown by Counter-Example \ref{Morse}. It follows that this category is not a [[reflective subcategory]] of $Top$, as $Haus$ is. \begin{remark} \label{retracts}\hypertarget{retracts}{} A small saving grace is that the category of normal spaces is [[Cauchy complete category|Cauchy complete]], in fact is closed under retracts in $Top$. This is so whether or not the Hausdorff condition is included. (If $r: Y \to X$ retracts $i: X \to Y$, then $r$ is a quotient map and $i$ is a subspace inclusion. If $A, B$ are closed and disjoint in $X$, then $r^{-1}(A), r^{-1}(B)$ are closed and disjoint in $Y$. Separate them by disjoint open sets $U \supseteq r^{-1}(A), V \supseteq r^{-1}(B)$ in $Y$; then $i^{-1}(U), i^{-1}(V)$ are disjoint open sets separating $i^{-1} r^{-1}(A) = A, i^{-1} r^{-1}(B) = B$ in $X$.) \end{remark} More at the page \emph{[[colimits of normal spaces]]}. \hypertarget{references}{}\subsection*{{References}}\label{references} The class of normal spaces was introduced by Tietze (1923) and Aleksandrov--Uryson (1924). \begin{itemize}% \item Ryszard Engelking, \emph{General topology}, (Monographie Matematyczne, tom 60) Warszawa 1977; expanded Russian edition Mir 1986. \item [[Misha Gavrilovich]], \emph{Point set topology as diagram chasing computations}, (\href{https://arxiv.org/abs/1408.6710}{arXiv:1408.6710}). \end{itemize} [[!redirects normal space]] [[!redirects normal spaces]] [[!redirects Normal space]] [[!redirects Normal spaces]] [[!redirects normal topological space]] [[!redirects normal topological spaces]] [[!redirects T4 space]] [[!redirects T4 spaces]] [[!redirects T4-space]] [[!redirects T4-spaces]] [[!redirects T-4 space]] [[!redirects T-4 spaces]] [[!redirects T-4-space]] [[!redirects T-4-spaces]] [[!redirects normal Hausdorff space]] [[!redirects normal Hausdorff spaces]] [[!redirects normal Hausdorff topological space]] [[!redirects normal Hausdorff topological spaces]] [[!redirects R3 space]] [[!redirects R3 spaces]] [[!redirects R3-space]] [[!redirects R3-spaces]] [[!redirects R-3 space]] [[!redirects R-3 spaces]] [[!redirects R-3-space]] [[!redirects R-3-spaces]] [[!redirects normal regular space]] [[!redirects normal regular spaces]] [[!redirects normal locale]] [[!redirects normal locales]] \end{document}