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compact_topological_spaces [2010/07/29 15:46] (current)
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 +=====Compact topological spaces=====
 +
 +Abbreviation: **KTop**
 +
 +====Definition====
 +A \emph{compact topological space} is a [[topological space]] $\mathbf{X}=\langle X,\Omega\rangle$ that is
 +
 +\emph{compact}: every open cover has a finite subcover, i.e.,
 +$\forall\mathcal C\subseteq\Omega(\bigcup\mathcal C=X\Longrightarrow\exists n, \exists C_0,\ldots,C_{n-1}\in\mathcal C(C_0\cup\cdots\cup C_{n-1}=X))$
 +
 +Remark: This is a template.
 +If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.
 +
 +It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
 +
 +==Morphisms==
 +Let $\mathbf{X}$ and $\mathbf{Y}$ be compact topological spaces. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:X\rightarrow Y$ that is a continuous:
 +$\forall V\in\Omega_{\mathbf Y}(h^{-1}[Y]\in\Omega_{\mathbf X})$
 +
 +====Definition====
 +A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
 +...\rangle$ such that
 +
 +$...$ is ...:  $axiom$
 +  
 +$...$ is ...:  $axiom$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
 +
 +^[[Classtype]]                        |second-order  |
 +^[[Amalgamation property]]            | |
 +^[[Strong amalgamation property]]     | |
 +^[[Epimorphisms are surjective]]      | |
 +
 +====Subclasses====
 +  [[Compact Hausdorff topological spaces]]
 +
 +
 +====Superclasses====
 +  [[Topological spaces]]
 +
 +
 +====References====
 +
 +[(Lastname19xx>
 +F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
 +)]
 +
 +