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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]] | ||

+ | )] | ||

+ | |||

+ | |||

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