# Differences

This shows you the differences between two versions of the page.

hausdorff_spaces [2010/07/29 15:46] (current)
Line 1: Line 1:
+=====Hausdorff spaces=====
+
+Abbreviation: **Haus**
+
+====Definition====
+A \emph{Hausdorff space} or \emph{$T_2$-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 disjoint open sets containing each point:  $x,y\in X\Longrightarrow\exists U,V\in\Omega(\mathbf{X})[x\in U\mbox{ and }y\in V\mbox{ and }U\cap V=\emptyset]$
+
+==Morphisms==
+Let $\mathbf{X}$ and $\mathbf{Y}$ be Hausdorff spaces.
+A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $f:X\rightarrow Y$ that is \emph{continuous}:
+
+$V\in\Omega(\mathbf{Y})\Longrightarrow f^{-1}[V]\in\Omega(\mathbf{X})$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+
+====Properties====
+^[[Classtype]]  |second-order |
+^[[Amalgamation property]]  |no |
+^[[Strong amalgamation property]]  |no |
+^[[Epimorphisms are surjective]]  |no |
+
+Remark:
+The properties given above use an $(\mathcal{E},\mathcal{M})$ factorization system with $\mathcal{E}=$ surjective morphisms and
+$\mathcal{M}=$ embeddings.
+
+
+
+====Subclasses====
+[[Compact Hausdorff spaces]]
+
+[[Completely Hausdorff spaces]]
+
+
+====Superclasses====
+[[T1-spaces]]
+
+
+