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Encyclopedia of Compactness Wiki*
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compact をテンプレートにして作成
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開始行:
*Definition [#d1a56521]
A topological space is called compact if every open cover has a finite subcover.
*Property [#m5c34cf9]
-In Hausdorff space, every compact subset is closed.
-Every closed subspace of compact space is compact.
-Compactness is equivalent to each of the following properties.
++Every net has a convergent subnet(the Borzano-Weierstrass theorem).
++Every closed family with [[fip]] has nonempty intersection.
++There are no [[maximal]] open covers.
-Every compact Hausdorff space is normal.
-For a Hausdorff space X the following conditions are equivalent:
++The space X is compact.
++For every topological space Y the projection p:X×Y→Y is closed.
++For every normal space Y the projection p:X×Y→Y is closed.
*Remark [#x1f778db]
-Some authors, like Bourbaki, require compact spaces to be Hausdorff. cf. [[quasicompact]]
*Reference [#o6ee5652]
Kelley, ''General Topology'', Springer (1975)
終了行:
*Definition [#d1a56521]
A topological space is called compact if every open cover has a finite subcover.
*Property [#m5c34cf9]
-In Hausdorff space, every compact subset is closed.
-Every closed subspace of compact space is compact.
-Compactness is equivalent to each of the following properties.
++Every net has a convergent subnet(the Borzano-Weierstrass theorem).
++Every closed family with [[fip]] has nonempty intersection.
++There are no [[maximal]] open covers.
-Every compact Hausdorff space is normal.
-For a Hausdorff space X the following conditions are equivalent:
++The space X is compact.
++For every topological space Y the projection p:X×Y→Y is closed.
++For every normal space Y the projection p:X×Y→Y is closed.
*Remark [#x1f778db]
-Some authors, like Bourbaki, require compact spaces to be Hausdorff. cf. [[quasicompact]]
*Reference [#o6ee5652]
Kelley, ''General Topology'', Springer (1975)
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