All Questions
Tagged with gn.general-topology reference-request
325 questions
3
votes
0
answers
251
views
What is the origin of the metrization problem for compact convex sets?
The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...
- 3,547
13
votes
1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
24
votes
2
answers
4k
views
complement of a totally disconnected closed set in the plane
While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
16
votes
2
answers
4k
views
Is there a "disjoint union" sigma algebra?
I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally:
For an indexed family of sets $\{A_i\...
- 629
34
votes
4
answers
9k
views
Why are the integers with the cofinite topology not path-connected?
An apparently elementary question that bugs me for quite some time:
(1) Why are the integers with the cofinite topology not path-connected?
Recall that the open sets in the cofinite topology on a ...
- 5,743
16
votes
2
answers
4k
views
Is the space of continuous functions from a compact metric space into a Polish space Polish?
Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup_{k\in K}\ ...
user6096
13
votes
1
answer
719
views
Homotopy theory for spanning trees of a graph
I am studying a paper of L. Lovász, ``A homology theory for spanning trees of a graph,'' but professor Babai has told me that Lovász later realized that this work is better framed in the language of ...
- 5,898
7
votes
2
answers
2k
views
Every real-valued continuous function on a closed set of compact Hausdorff space has an extension.
I've noted, that the following fact can be proven in a few lines using $C^*$-algebra theory. I wonder if it has a simple elementary proof or not. Probably you can give me a reference.
Suppose $X$ ...
- 1,284
9
votes
3
answers
990
views
Is there a list of all connected T_0-spaces with 5 points?
Is there some place (on the internet or elsewhere) where I can find the number and preferably a list of all (isomorphism classes of) finite connected $T_0$-spaces with, say, 5 points?
In know that a $...
- 3,174
8
votes
2
answers
2k
views
End point compactification for metric spaces
Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...
- 3,033
24
votes
5
answers
8k
views
totally disconnected and zero-dimensional spaces
When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: covering dimension, small ...
- 3,605
7
votes
3
answers
590
views
Expressing any f(x,y) using only addition and unary functions?
Suppose we have a continuous function $f:R^2\rightarrow R$. I was told of the following remarkable theorem: $f$ can be expressed as the composition of continuous unary functions (that is, functions ...
- 3,979
1
vote
1
answer
908
views
What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...
10
votes
1
answer
2k
views
Different forms of compactness and their relation
Given a topological space X one can define several notion of compactness:
X is compact if every open cover has a finite subcover.
X is sequentially compact if every sequence has a convergent ...
- 3,004
24
votes
6
answers
5k
views
A good place to read about uniform spaces
I'd like to learn a bit about uniform spaces, why are they useful, how do they arise, what do they generalize, etc., without getting away from the context of general topology. I have to prepare an ...
- 3,004
9
votes
1
answer
1k
views
When completion of locally compact length space is locally compact?
As far as I know the answer to the question:
"Is it true that a completion of a locally compact length space is locally compact?" - Negative.
Does anybody know some metric and/or topological ...
- 141
33
votes
4
answers
7k
views
Topology of function spaces?
Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,...
- 33.3k
3
votes
1
answer
321
views
Removing intersections of curves in surfaces
Let $C_1, \dots, C_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of $\...
- 32.1k
6
votes
3
answers
582
views
profinite spaces coming from profinite groups
This is probably well-known:
Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed?
- Is every profinite group ...
- 63.1k
3
votes
1
answer
587
views
Functoriality of base change
Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism $\kappa$...
- 1,457
8
votes
4
answers
6k
views
Connectedness and the real line
It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be ...
- 3,830
10
votes
4
answers
5k
views
What is a reference for profinite sets?
The question is in the title. The motivation behind the question is as follows: there are plenty of references about profinite groups and profinite completions of groups. It seems that their not ...
- 207
36
votes
4
answers
4k
views
How far is Lindelöf from compactness?
A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the ...
- 2,396
26
votes
15
answers
19k
views
Learning Topology
EDIT (Harry): Since this question in its original form was poorly stated (asked about topology rather than graph theory), but we have a list of Topology books in the answers, I guess you should go ...
17
votes
10
answers
3k
views
References for homotopy colimit
(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
- 12.8k