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t1-spaces [2010/07/29 15:46] (current)
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+=====T1-spaces=====
+
+Abbreviation: **Top$_1$**
+
+====Definition====
+A \emph{$T_1$-space} is a [[topological spaces]] $\mathbf{X}=\langle X,\Omega(\mathbf{X})\rangle$ such that
+
+
+for every pair of distinct points in the space, there is a pair of open sets containing each point but not the other:  $x,y\in X\Longrightarrow\exists U,V\in\Omega(\mathbf{X})[x\in U\setminus V\mbox{ and }y\in V\setminus U]$
+
+==Morphisms==
+Let $\mathbf{X}$ and $\mathbf{Y}$ be $T_1$-spaces.
+A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $f:X\rightarrow Y$ that is \emph{continuous}:
+
+$V\in\Omega(\mathbf{Y})\Longrightarrow f^{-1}[V]\in\Omega(\mathbf{X})$
+
+====Definition====
+A \emph{$T_1$-space} is a [[topological spaces]] $\mathbf{X}=\langle X,\Omega(\mathbf{X})\rangle$ such that all
+
+singleton subsets are closed:  $X\setminus\{x\}\in\Omega(\mathbf{X})$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[Classtype]]  |second-order |
+^[[Amalgamation property]]  |yes |
+^[[Strong amalgamation property]]  |yes |
+^[[Epimorphisms are surjective]]  |yes |
+
+Remark:
+The properties given above use an $(\mathcal{E},\mathcal{M})$ factorization system with $\mathcal{E}=$ surjective morphisms and
+$\mathcal{M}=$ embeddings.
+
+
+
+====Subclasses====
+[[Hausdorff spaces]]
+
+
+====Superclasses====
+[[T0-spaces]]
+
+
+
+see also http://www.wikipedia.org/wiki/Topology_glossary
+
+
+====References====
+
+[(Ln19xx>
+)]\end{document}
+%</pre>