| 11. | ;Countably compact : A space is countably compact if every countable open cover has a finite subcover.
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| 12. | Use Zorn's Lemma to find an open cover without finite subcover that is " maximal " amongst such covers.
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| 13. | This area of north-west New South Wales, the Sand Plain Mulga Shrublands, supports an open cover of shrubs and tussock grasses.
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| 14. | A Hausdorff space X \, is paracompact if and only if it every open cover admits a subordinate partition of unity.
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| 15. | Formally, a topological space " X " is called " compact " if each of its open covers has a finite subcover.
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| 16. | The two-dimensional surface of a sphere S ^ 2 has an open cover by two contractible sets, open neighborhoods of opposite hemispheres.
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| 17. | Since these form an open cover for " X " and simplices are chain homotopic to the identity map on homology ).
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| 18. | Then there exists an infinite open cover " C " of " T " 0 that does not admit any finite subcover.
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| 19. | These neighborhoods consist of an open cover of the interval, and since the interval is compact there is a finite subcover of them.
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| 20. | One array inside the open cover and another on top of a stack of four others are exposed to solar winds at all times.
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