Differences

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

t1-spaces [2010/07/29 15:46] (current)
Line 1: Line 1:
 +=====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>