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I e each point of the space is.
Compact space. Any interval of the form with both and real numbers is a compact space with the subspace topology inherited from the usual topology on the real line. So the definitions are consistent. Compact subsets could look very different from unions of intervals.
A topological space x x x is compact if and only if it is compact as a subset of itself. A non empty topological space is compact in the original sense of the word or countably compact as they are now called if it satisfies any one of the following equivalent statements. An open covering of a space or set is a collection of open sets that covers the space.
Compact set heine borel theorem paracompact space topological space cite this as. More generally any finite union of such intervals is compact. It is not hard to show that z x z subseteq x z x is compact as a subset of x x x if and only if it is compact as a topological space when given the subspace topology.
If x is not hausdorff then the closure. Properties of compact spaces a compact subset of a hausdorff space x is closed. For instance the cantor setis compact.
The concept of a compact space was originally a strengthening of that of a compact space introduced by m. If x is not hausdorff then a compact subset of x may fail to be a. 3 the intersection.
If x is not hausdorff then a compact subset of x may fail to be a closed subset of x see footnote for example. This definition is often extended to the whole space. Compactness in mathematics property of some topological spaces a generalization of euclidean space that has its main use in the study of functions defined on such spaces.
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