All Questions
Tagged with compactness gn.general-topology
11 questions
48
votes
19
answers
17k
views
What is your favorite proof of Tychonoff's Theorem?
Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by Loomis:
https://archive.org/details/introductiontoab031610mbp
https://ia800309.us.archive....
11
votes
2
answers
314
views
Spaces with every compactification $0$-dimensional which aren't locally compact
Recently I've proven the following theorem
Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent:
Every compactification of $X$ is zero-dimensional....
- 1,201
7
votes
1
answer
899
views
Is a closed subset of an extremally disconnected set again extremally disconnected?
Let $T$ be a compact Hausdorff extremally disconnected set (so $T$ is a compact Hausdorff space, such that the closure of each open subset is again open). Let $S \subseteq T$ be a closed subset.
...
- 755
32
votes
3
answers
6k
views
Is "compact implies sequentially compact" consistent with ZF?
Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
- 26.8k
13
votes
1
answer
602
views
A generalization of the Arhangelskii Theorem
Arhangeleskii's Theorem states the following
For any Hausdorff topological space $X$,
$$
|X|\leq2^{\chi(X)L(X)}
$$
where $\chi(X)$ is the character of $X$ and $L(X)$ is the Lindelöf degree of $...
Ottenbreit
12
votes
1
answer
2k
views
What are compact objects in the category of topological spaces?
Let $\mathscr C$ be a locally small category that has filtered colimits. Then an object $X$ in $\mathscr C$ is compact if $\operatorname{Hom}(X,-)$ commutes with filtered colimits.
On the other hand, ...
- 28.7k
11
votes
1
answer
309
views
Which closed subsets $Y$ of a compact space $X$ admit a linear extensor $C(Y)\to C(X)$?
In the following $X$ is a Hausdorff compact topological space. Let $Y$ be a closed subset of $X$.
The restriction operator $R_Y:C(X)\to C(Y)$ is surjective (Tietze), so it admits a continuous right ...
- 60.5k
8
votes
2
answers
579
views
Totally disconnected subspaces
This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then.
Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it ...
- 81
6
votes
2
answers
2k
views
Is there an easier proof to show that the closed convex hull of a normalized weakly null sequence is weakly compact?
In a paper that I am reading there is a following step:
Let $X$ be a Banach space and let $(x_k) \subset X$ be a normalized sequence that converges weakly to $0$.
Then $\overline{co}(x_k)$ is a ...
- 163
2
votes
1
answer
103
views
LCH spaces $X$ such that if $Y$ is a perfect image of $X$, then $Y$ is zero-dimensional
I am looking for locally compact Hausdorff spaces $X$ with the following property:
If $f:X\to Y$ is a perfect map onto locally compact Hausdorff space $Y$, then $Y$ is zero-dimensional.
One can see ...
- 1,201
2
votes
3
answers
435
views
Compact, densely ordered spaces
During my work with order preserving homeomorphisms, I got interested in the double arrow space and, subsequently, in the lexicographic square.
I would really like to find examples of spaces like ...
- 203