* The Cantor set. This is a corollary of the previous example and Tychonoff's Theorem (see below). |
* The Cantor set. |
* A subset of Euclidean n-space is compact if and only if it is closed and bounded. (Heine-Borel Theorem) |
* A subset of Euclidean n-space is compact if and only if it is closed and bounded. (Heine-Borel theorem) |
* A topological space is compact if and only if every filter on the space has a convergent refinement. |
* Pseudocompact: Every real-valued function on the space is bounded. |
* Pseudocompact: Every real-valued continuous function on the space is bounded. |
Compact spaces are countably compact, as are sequentially compact spaces. Countably compact spaces are pseudocompact and weakly countably compact. |
While all these concepts are equivalent for metric spaces, in general we have the following implications: Compact spaces are countably compact. Sequentially compact spaces are countably compact. Countably compact spaces are pseudocompact and weakly countably compact. /Talk |
Some examples of compact spaces:
Some theorems related to compactness:
There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. These include the following.
While all these concepts are equivalent for metric spaces, in general we have the following implications: Compact spaces are countably compact. Sequentially compact spaces are countably compact. Countably compact spaces are pseudocompact and weakly countably compact.