All Questions
Tagged with compactness ct.category-theory
7 questions
7
votes
1
answer
134
views
Universally closed implies proper for locales
It is well known that:
Theorem.
For a locale (resp. topological space) $X$, the following are equivalent:
$X$ is compact, i.e. every open cover of $X$ has a finite subcover.
For every locale (resp. ...
- 15.9k
6
votes
0
answers
246
views
Making the analogy of finiteness and compactness precise
If one asks about the intution behind compact topological spaces, most often one will hear the mantra
“Compactness of a topological space is a generalisation of the finiteness of a set.”
For example,...
- 1,474
10
votes
1
answer
448
views
Do compactly generated spaces have a more direct definition?
Is there an elementary way to define Haussdorf-compactly generated weakly Hausdorff topological spaces in a way that does not need defining topological space first?
Weakly Hausdorff sequential spaces ...
- 1,947
7
votes
2
answers
1k
views
CG spaces from the perspective of sheaves over compact Hausdorff spaces
A compactly generated space is a space $X$ such that $f : X \rightarrow Y$ is continuous if and only if $K \rightarrow X \stackrel{f}{\rightarrow} Y$ is continuous for each compact hausdorff space $K$....
user30211
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
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
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