t1-spaces - MathStructures

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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====

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