# Differences

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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]]
+)]
+
+