History of Compact space

	Revision 23 . . (edit) December 16, 2001 2:13 am by AxelBoldt
	Revision 22 . . (edit) December 14, 2001 5:20 pm by Zundark [correct something that used to be correct but got changed]
	Revision 21 . . December 1, 2001 3:57 am by AxelBoldt
	Revision 20 . . December 1, 2001 3:56 am by Zundark [link Heine-Borel Theorem]
	Revision 19 . . (edit) September 30, 2001 11:47 pm by Zundark [fix formatting]

---

Difference (from prior major revision) (minor diff, author diff)

Changed: 13c13

* The Cantor set. This is a corollary of the previous example and Tychonoff's Theorem (see below).

* The Cantor set.

Changed: 25c25

* 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)

Added: 34a35,36

* A topological space is compact if and only if every filter on the space has a convergent refinement.

Removed: 47,48d48

* Pseudocompact: Every real-valued function on the space is bounded.

Added: 50a51,52

* Pseudocompact: Every real-valued continuous function on the space is bounded.

Changed: 53c55,61

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?