separation axioms SeparationAxioms 2003-02-23 14:57:19 2007-05-27 00:42:56 Definition Hausdorff completely Hausdorff normal completely normal regular Tychonoff completely regular perfectly normal Tychonov perfectly $T_4$ % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \renewcommand{\text}{\textnormal} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \PMlinkescapeword{name} \PMlinkescapeword{names} \PMlinkescapeword{type} \PMlinkescapeword{types} \PMlinkescapeword{order} \PMlinkescapeword{axiom} The \emph{separation axioms} are additional conditions which may be required to a topological space in order to ensure that some particular types of sets can be separated by open sets, thus avoiding certain pathological cases. \begin{center} \begin{tabular*}{.8\textwidth}{lp{0.65\textwidth}} \textbf{Axiom} & \textbf{Definition} \\ \hline $T_0$ & given two distinct points, there is an open set containing exactly one of them; \\ \PMlinkname{$T_1$}{T1Space} & given two distinct points, there is a neighborhood of each of them which does not contain the other point;\\ \PMlinkname{$T_2$}{T2Space} & given two distinct points, there are two disjoint open sets each of which contains one of the points;\\ $T_{2\frac{1}{2}}$ & given two distinct points, there are two open sets, each of which contains one of the points, whose closures are disjoint;\\ \PMlinkname{$T_3$}{T3Space} & given a closed set $A$ and a point $x\notin A$, there are two disjoint open sets $U$ and $V$ such that $x\in U$ and $A\subset V$;\\ $T_{3\frac{1}{2}}$ & given a closed set $A$ and a point $x\notin A$, there is an Urysohn function for $A$ and $\{b\}$;\\ $T_4$ & given two disjoint closed sets $A$ and $B$, there are two disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$;\\ $T_5$ & given two separated sets $A$ and $B$, there are two disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$.\\ \end{tabular*} \end{center} If a topological space satisfies a $T_i$ axiom, it is called a $T_i$-space. The following table shows other common names for topological spaces with these or other additional separation properties. \begin{center} \begin{tabular}{ll} \textbf{Name} & \textbf{Separation properties} \\ \hline Kolmogorov space & $T_0$ \\ Fr\'echet space & $T_1$ \\ Hausdorff space & $T_2$ \\ Completely Hausdorff space & $T_{2\frac{1}{2}}$\\ Regular space & $T_3$ and $T_0$\\ Tychonoff or completely regular space & $T_{3\frac{1}{2}}$ and $T_0$\\ Normal space & $T_4$ and $T_1$\\ Perfectly $T_4$ space & $T_4$ and every closed set is a $G_\delta$ (see \PMlinkname{here}{G_deltaSet}) \\ Perfectly normal space & $T_1$ and perfectly $T_4$\\ Completely normal space & $T_5$ and $T_1$\\ \end{tabular} \end{center} The following implications hold strictly: \begin{align*} (T_2 \text{ and } T_3) &\Rightarrow T_{2\frac{1}{2}}\\ (T_3 \text{ and } T_4) &\Rightarrow T_{3\frac{1}{2}}\\ T_{3\frac{1}{2}} &\Rightarrow T_3\\ T_5 &\Rightarrow T_4 \end{align*} \begin{align*} \text{Completely normal } &\Rightarrow \text{ normal }\Rightarrow \text{ completely regular }\\ &\Rightarrow \text{ regular }\Rightarrow T_{2\frac{1}{2}}\Rightarrow T_2\Rightarrow T_1 \Rightarrow T_0 \end{align*} \textbf{Remark.} Some authors define $T_3$ spaces in the way we defined regular spaces, and $T_4$ spaces in the way we defined normal spaces (and vice-versa); there is no consensus on this issue. \medskip \textbf{Bibliography:} \emph{Counterexamples in Topology}, L. A. Steen, J. A. Seebach Jr., Dover Publications Inc. (New York)