Compact Space

A subset of a topological space is compact if it is compact as a topological space with the relative topology i e every family of open sets of whose union contains has a finite subfamily whose union contains.

Compact space. 1 each countable open covering of subsets of this space contains a finite subcovering. If x is not hausdorff then a compact subset of x may fail to be a. A topological space x x x is compact if and only if it is compact as a subset of itself. So the definitions are consistent.

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. 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. For instance the cantor setis compact. An open covering of a space or set is a collection of open sets that covers the space.

The concept of a compact space was originally a strengthening of that of a compact space introduced by m. More generally any finite union of such intervals is compact. Compact set heine borel theorem paracompact space topological space cite this as. 3 the intersection.

Compact subsets could look very different from unions of intervals. I e each point of the space is. If x is not hausdorff then the closure. 2 the intersection of any countable centred system of non empty subsets is non empty.

This definition is often extended to the whole space. 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.