All Questions
Tagged with compactness gn.general-topology
80 questions
10
votes
2
answers
797
views
On the definition of locally compact for non-Hausdorff spaces
It seems that there are different conventions in the literature as to what is a locally compact space (when the space is not supposed Hausdorff).
The two main non equivalent definitions I've seen ...
- 1,035
13
votes
1
answer
1k
views
A topology on $\Bbb R$ where the compact sets are precisely the countable sets
QUESTION.
In there a topology on $\Bbb R$ where the compact subsets are precisely the countable subsets?
I am trying to create a counterexample to a certain claim, and I found that what I need is a ...
- 233
8
votes
3
answers
612
views
Is there a non-metrizable topological space for which any countably compact subset is compact?
The title is the question : Is there a non-metrizable topological space for which any countably compact subset is compact ?
EDIT : non-metrizable and Hausdorff
- 361
6
votes
2
answers
2k
views
How do I prove that compact-open topology is metrizable?
Let $X$ be a $\sigma$-compact topological space and $(Y,d)$ be a metric space.
Let $\{K_n\}$ be a sequence of compact subsets of $X$ whose union is $X$.
Define $\rho_n(f,g):=\sup \{d(f(z),g(z)): z\...
- 337
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
1
vote
0
answers
127
views
Category-theoretic characterization of zero-dimensional spaces
Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
- 369
2
votes
1
answer
211
views
A quasicompact space with a net that contains no convergent strict subnet
If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...
- 1,270
2
votes
0
answers
96
views
On compactness in $C(X)$
Let $X$ be a Tychonoff space. It is well known, that for a family of scalar functions equicontinuity + pointwise boundedness imply relative compactness in $C(X)$ (with compact-open topology). It is ...
- 5,529
3
votes
0
answers
201
views
Which compact topological spaces are homeomorphic to their ultrapower?
It is well known that for any compact metric space $(X, d)$, and any ultrafilter $\mu$ there is a map $i_\mu:\prod_\mu (X, d) \to (X_d)$ in the category of metric spaces and Lipschitz maps where $i_\...
- 2,221
3
votes
1
answer
970
views
On compactness in Sion's minimax theorem
Sions minimax theorem (wiki, paper) can be stated as follows:
Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex
subset of a linear topological space. Let $f$ be a ...
- 359
2
votes
0
answers
206
views
Regularity of Dirac measure on Baire sets [closed]
Suppose $X$ is a locally compact Hausdorff space.
Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$,
to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$.
...
- 373
5
votes
2
answers
575
views
Anti-compactness
Let $(X,\tau)$ be a topological space such that
$\tau\ne\{\emptyset\ X\}.\ $
We call an open cover $\mathcal{U}$ of $(X,\tau)$ proper if
$\ X\notin \mathcal{U}.\ $ Moreover we say that $(X,\...
- 48.9k
2
votes
1
answer
331
views
Relative Compactness vs Way Below in Locally Compact Hausdorff Spaces
Let $Y$ be a subset of a locally compact Hausdorff topological space $X$ and consider the following properties.
$\overline{Y}$ is compact.
Every open cover of $X$ has a finite subcover of $Y$.
...
- 1,307
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
3
votes
2
answers
749
views
History of limit point compact -/-> compact example
A standard example in elementary topology (e.g. Munkres) of a space that is limit-point compact (every infinite subset of the space has a limit point) but not compact is the minimal uncountable well-...
- 339
2
votes
2
answers
346
views
Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?
Definitions and notations.
Let $\mathcal{P}(X)$ the power set of $X$.
Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X.
We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...
- 163
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
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
5
votes
2
answers
638
views
Beyond Cantor's Teepee
From Counterexamples in Topology by Steen and Seebach (2nd edition) example 129 page 145 we have an example of connected and totally path-disconnected space.
It is defined as follow:
Fix $p= (1/2,1/2)...
- 2,829
2
votes
3
answers
465
views
Axiomatization of locally compact Hausdorff spaces via compact subspaces
The usual axiomatization of a topological space (in the sense of Bourbaki) goes by declaring certain subsets as being open and such that a few axioms are fulfilled by the family of open subsets.
It ...
- 2,399
27
votes
1
answer
840
views
Can closed compacts in a topological group behave "paradoxically" with respect to unions, intersections, and one-sided translations?
Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$:
$A' = aA$,
$B' = bB$.
Suppose it is known that $A'\...
- 1,481
2
votes
1
answer
226
views
A question on countably compact space
A regular space $X$ is
star compact (which implies pseudocompact)
with $G_\delta$-diagonal
star countable
first countable
$e(X)\le \aleph_0$ ( in fact it implies star countable)
$|X|=\aleph_1$
Cech-...
- 654
3
votes
3
answers
895
views
Compactness of sigma-algebra for the $L^1$ metrics
Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metric $d(A,B) = \mu(A \triangle B)$. Is $(G,d)$ a compact metric space?
...
- 5,721
3
votes
1
answer
418
views
Compact subsets and Hausdorffness of topology
We know that all closed subsets of a compact topological space $(X,\tau)$ are compact. If we add the Hausdorff condition on the topology $\tau$ we can see the equivalence of these conditions on a ...
- 1,788
5
votes
2
answers
3k
views
Zariski topology and compact \paracompact space?
Does the Zariski topology on a ring (not commutative in common) form a compact or paracompact space and why?
- 53
1
vote
3
answers
688
views
How to show the cardinality of nonisometric compact metric spaces is the continuum
It is asserted in A Course in Metric Geometry by Burago, Burago, Ivanov that
there can be no more than continuum of mutually nonisometric compact spaces
How is this proven?
Its clear that there ...
- 7,197
6
votes
14
answers
5k
views
Applications of compactness [closed]
Similar to this question: Applications of connectedness I want to collect applications of compactness.
E.g.: compact + discrete => finite, which can be used to prove the finiteness of the ...
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....
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
17
votes
5
answers
830
views
How can one characterise compactness-by-experiment?
There are a myriad different variations on the theme of "compactness", and some of them have even made it on to Wikipedia. I'm interested in finding out more about types of compactness that ...
- 26.8k