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