Countably compact space

In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.

Equivalent definitions edit

A topological space X is called countably compact if it satisfies any of the following equivalent conditions: [1][2]

(1) Every countable open cover of X has a finite subcover.
(2) Every infinite set A in X has an ω-accumulation point in X.
(3) Every sequence in X has an accumulation point in X.
(4) Every countable family of closed subsets of X with an empty intersection has a finite subfamily with an empty intersection.
Proof of equivalence

(1)   (2): Suppose (1) holds and A is an infinite subset of X without  -accumulation point. By taking a subset of A if necessary, we can assume that A is countable. Every   has an open neighbourhood   such that   is finite (possibly empty), since x is not an ω-accumulation point. For every finite subset F of A define  . Every   is a subset of one of the  , so the   cover X. Since there are countably many of them, the   form a countable open cover of X. But every   intersect A in a finite subset (namely F), so finitely many of them cannot cover A, let alone X. This contradiction proves (2).

(2)   (3): Suppose (2) holds, and let   be a sequence in X. If the sequence has a value x that occurs infinitely many times, that value is an accumulation point of the sequence. Otherwise, every value in the sequence occurs only finitely many times and the set   is infinite and so has an ω-accumulation point x. That x is then an accumulation point of the sequence, as is easily checked.

(3)   (1): Suppose (3) holds and   is a countable open cover without a finite subcover. Then for each   we can choose a point   that is not in  . The sequence   has an accumulation point x and that x is in some  . But then   is a neighborhood of x that does not contain any of the   with  , so x is not an accumulation point of the sequence after all. This contradiction proves (1).

(4)   (1): Conditions (1) and (4) are easily seen to be equivalent by taking complements.

Examples edit

Properties edit

See also edit

Notes edit

  1. ^ Steen & Seebach, p. 19
  2. ^ "General topology - Does sequential compactness imply countable compactness?".
  3. ^ Steen & Seebach 1995, example 42, p. 68.
  4. ^ Steen & Seebach, p. 20
  5. ^ Steen & Seebach, Example 105, p, 125
  6. ^ Willard, problem 17G, p. 125
  7. ^ Kremsater, Terry Philip (1972), Sequential space methods (Thesis), University of British Columbia, doi:10.14288/1.0080490, Theorem 1.20
  8. ^ Willard, problem 17F, p. 125
  9. ^ Willard, problem 17F, p. 125
  10. ^ a b "General topology - Countably compact paracompact space is compact".
  11. ^ Steen & Seebach, Figure 7, p. 25
  12. ^ "General topology - Prove that a countably compact, first countable T2 space is regular".
  13. ^ Willard, problem 17F, p. 125
  14. ^ "General topology - is the Product of a Compact Space and a Countably Compact Space Countably Compact?".
  15. ^ Engelking, example 3.10.19, p. 205

References edit