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Results tagged with compactness
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user 5903
5
votes
A question about locally compact spaces
See also Compactness and product spaces, Coll. Math., 7 (1959), 19--22 by S. Mrowka. …
- 11.4k
3
votes
Accepted
Is the Čech–Stone compactification of the integers always a retract of an extremally disconn...
$\beta\mathbb{N}$ is not a retract of a Tychonoff cube because it is not connected; it also not a retract of a Cantor cube, not even a continuous image, see problem 3.12.12 in Engelking's book.
It is …
- 11.4k
15
votes
Is "compact implies sequentially compact" consistent with ZF?
The sequential compactness of $[0,1]^{\omega_1}$ is undecidable in ZFC: as noted above $[0,1]^{[0,1]}$ is not, so under CH $[0,1]^{\omega_1}$is not sequentially compact; on the other hand $\mathrm{MA}+ …
- 11.4k
3
votes
Accepted
LCH spaces $X$ such that if $Y$ is a perfect image of $X$, then $Y$ is zero-dimensional
Using @Anonymous' second comment one can show that your property characterizes locally compact scattered spaces.
For if $X$ is locally compact Hausdorff and not scattered then it contains a closed den …
- 11.4k
2
votes
Extending homeomorphisms between compact metric subsets
As you can see from the comments the answer is: hardly ever.
As mentioned above the case of one-point sets necessitates the space being homogeneous. But that is not enough, say in $\mathbb{R}$ when y …
- 11.4k
10
votes
Accepted
Characterization of pretty compact spaces
A partial answer: other examples of pretty compact spaces are uncountable powers of $\{0,1\}$ and $[0,1]$, and in general products of uncountably many non-trivial compact Hausdorff spaces. See Problem …
- 11.4k
13
votes
What is your favorite proof of Tychonoff's Theorem?
One proves this theorem word for word as in he case of the compactness of the product of intervals". …