Every sequentially compact space is countably compact. Conversely, every first countable countably compact space is sequentially compact. The ordinal space is sequentially compact but not first countable, since has not countable local basis.
Next, compactness and sequential compactness are not compatible. In other words, neither one implies the other. Here’s an example of a compact space that is not sequentially compact. Let , where is the closed unit interval (with the usual topology), and is equipped with the product topology. Then is compact (since is, together with Tychonoff theorem). However, is not sequentially compact. To see this, let be the function such that for any , is the -th digit of in its binary expansion. But the sequence has no convergent subsequences: if is a subsequence, let such that its binary expansion has its -th digit iff is odd, and otherwise. Then is the sequence , and is clearly not convergent. The ordinal space is an example of a sequentially compact space that is not compact, since the cover has no finite subcover.
is sequentially compact.
|Date of creation||2013-03-22 12:50:05|
|Last modified on||2013-03-22 12:50:05|
|Last modified by||mps (409)|