All Questions
5,183 questions
3
votes
3
answers
4k
views
Compactness and Covering Spaces
Let p : Y -> X be an n-sheeted covering map, where X and Y are topological spaces. If X is compact, prove that Y is compact.
I realize that this seems like a very simple problem, but I want to stress ...
- 31
13
votes
11
answers
4k
views
Are nets and filters useful in geometry and topology?
Many results in topology can be restated using the concepts of nets and ultrafilters. This seems to be of interest for set theorists, maybe even logicians. But for geometers and topologists, those who ...
- 15.4k
-2
votes
2
answers
954
views
Three modifications of connectedness
This question arose in my research of generalized connectedness (see this draft article for the overall idea, but beware that the draft is yet too preliminary and unreadable, however I hope you can ...
-11
votes
1
answer
2k
views
Union of uniformly connected sets
I will call a set uniformly connected regarding some uniform space when it is connected regarding every entourage of this uniform space (entourages are considered as digraphs and it is taken strong ...
- 765
3
votes
3
answers
384
views
Collapsing contractible subsets of the two-disk.
This question is quite specific, but it may admit answers in more general contexts.
Consider a subset $\Lambda \subset D^2$ where $D^2$ is the two dimensional disk.
We consider in $\Lambda$ an ...
- 3,928
3
votes
2
answers
774
views
Question about closed projection
I'm wondering if the following can be true:
Let Y be a second countable space and
$\pi_2:Y \times \mathbb{R}\rightarrow\mathbb{R}$ ($\mathbb{R}$ with its usual topology and
$\pi_2$ the projection onto ...
- 1,727
1
vote
1
answer
387
views
Why not use usual topology in ordered spaces ?
This is a similar question to the one about the lack of use of usual topologies in measure theory. By usual topology here is meant the Hausdorff-Kuratowski-Bourbaki concept, based on open sets, or ...
2
votes
1
answer
2k
views
Why not usual topology in measure theory ?
Measure theory was introduced in the early 1900s by Lebesgue, at the same time with Hausdorff introducing the usual concept of topology, and publishing it in his book just before World War I. Measure ...
- 105
3
votes
1
answer
370
views
Weaker form of irreducible surjections
An irreducible surjection is usually defined as a continuous closed surjective map $f:X\rightarrow Y$ such that if for some closed set $C\subset X$ one has $f(C)=Y$ then $C=X$. In my dissertation I ...
- 2,275
10
votes
2
answers
1k
views
Is there a way to see a topological group as the "Cayley graph" of its "infinitesimal generators"?
At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...
- 54.6k
4
votes
0
answers
1k
views
Associative binary operations on natural numbers
Which are all the associative binary operations on natural numbers ?
Certain results in this regard can be found in arxiv:math/0508215.
It appears that such associative operations cannot grow too fast....
12
votes
1
answer
1k
views
Fixed point theorems and equiangular lines
I've been thinking about the equiangular lines (or SIC-POVM) conjecture, and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking ...
- 6,342
5
votes
1
answer
329
views
Example of a quasitopological group with discontinuous power map
A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion $G\...
- 7,193
5
votes
1
answer
976
views
How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?
How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...
- 1,499
9
votes
3
answers
1k
views
Relatively countably compact subsets without countably compact closure.
I am looking for a (Hausdorff or better) space $X$ and a subset $A$ of $X$ that is relatively countably compact (every sequence from $A$ has an accumulation point in $X$) such that the closure of $A$ ...
- 5,407
67
votes
10
answers
12k
views
Non-homeomorphic spaces that have continuous bijections between them
What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are not homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?
-1
votes
2
answers
466
views
Union of proximally connected sets
Let (δ;U) is a proximity space.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
Is the following true? (I need a proof or a counter-example.)
Conjecture If S ...
- 765
1
vote
3
answers
5k
views
Definition of Connected Subspace
In Munkres (Chapter 3, Section 23, p. 148), Munkres shows that if a subspace $Y$ of a space $X$ is not connected, then there are two disjoint open subsets $A,B$ such that the union of $A$ and $B$ ...
- 15.4k
9
votes
1
answer
3k
views
The Wedge Sum of path connected topological spaces
A definition of wedge sum can be found here:
http://en.wikipedia.org/wiki/Wedge_sum
My professor has claimed that wedge sums of path connected spaces X and Y are well-defined up to homotopy ...
- 95
3
votes
1
answer
2k
views
What is the pure intuition for topological continuity and topology? [closed]
I have read the introductory sections of many books on Real Analysis and Topology, yet nowhere have I found an unbiased motivation for the notions of either topology or (topological) continuity.
The ...
- 191
1
vote
1
answer
262
views
$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
- 1,882
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
3
answers
2k
views
motivation for compactness [duplicate]
Possible Duplicate:
How to understand the concept of compact space
Hello,
I am learning some analysis on my own and
what is the motivation to consider compactness?
eg. I do not understand why ...
- 317
-2
votes
1
answer
476
views
Countable open subgroup
In a Hausdorff topological group, how can I show that every infinite topological group has a countable open subgroup?
- 7
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
9
votes
1
answer
915
views
Is there a notion of a "perfectly regular" topological space?
The separation axioms have exploded a little since the original list of four! Amongst them can be found "completely regular" spaces and "perfectly normal" spaces. The former is well-known: a point ...
- 26.8k
3
votes
0
answers
126
views
More on continuous images of dense orders
In this question I asked if there was an analogue of connectedness which applied to dense orders which were not required to be complete. Between them, Noah and Joel showed that every (infinite) ...
- 3,612
9
votes
3
answers
1k
views
Structure Theorem for finitely generated commutative cancellative monoids?
Is there a Structure Theorem for finitely generated commutative cancellative monoids?
Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, ...
- 63.1k
5
votes
1
answer
232
views
When can boundedness be characterized topologically in Metric spaces?
Let H be a separable and infinite-dimensional Hilbert space. Is every closed subset of H homeomorphic
to some closed and bounded subset of H?
- 6,215
3
votes
2
answers
340
views
What, if anything, can be said about continuous images of densely ordered spaces?
If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove ...
- 3,612
5
votes
1
answer
296
views
Solenoid of a continuous map of a ball, is it contractible?
Let $B$ be the closed unit ball in $\mathbb R^n$ and $f\colon B\to B$ a continuous map.
Consider the infinite product $B^{\mathbb Z}$ equipped with the product topology. Consider the solenoid
$$
S_f=\...
- 4,188
0
votes
2
answers
505
views
Partition into connected sets by proximity
Let (δ;U) is a proximity space.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
I will call connected component a maximal connected set.
Is this true: U is ...
- 765
0
votes
1
answer
137
views
Connectedness of a union regading a proximity
Let δ is a proximity.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
Question: Let A and B are sets with non-empty intersection. Let both A and B ...
- 765
4
votes
2
answers
452
views
A family of subsets with a "gluing" property
Somewhat in line with this previous MathOverflow question:
I'm looking at a combinatorial structure consisting of a finite set $S$ of objects, and a family $F$ of designated subsets of $S$. We call ...
- 306
1
vote
4
answers
8k
views
Does Cauchy continuity imply uniform continuity? [No.] [closed]
It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff
$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...
- 3,830
17
votes
1
answer
3k
views
Do these conditions on a semigroup define a group?
As is well known, if $S$ is a semigroup in which the equations $a=bx$ and $a=yb$ have solutions for all $a$ and $b$, then $S$ is a group. This question arose when someone misunderstood the conditions ...
- 7,217
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 ...
5
votes
2
answers
709
views
profinite spaces are the pro-completion of finite sets
The title sounds tautological, right? Perhaps I'm missing something completely trivial here ...
Assume $X$ is a compact totally disconnected hausdorff space. It is known that $X$ can be written as ...
- 63.1k
16
votes
3
answers
8k
views
Defining Quotient Bundles
This is an extremely elementary question but I just can't seem to get things to work out. What I am looking for is a natural definition of the quotient bundle of a subbundle $E'\subset E$ of $\...
- 2,283
10
votes
1
answer
898
views
Category Theory / Topology Question
Let me begin by noting that I know quite little about category theory. So forgive me if the title is too vague, if the question is trivial, and if the question is written poorly.
Let $\mathcal{C}$ ...
- 29.2k
19
votes
3
answers
2k
views
How many tacks fit in the plane?
Call a tack the one point union of three open intervals. Can you fit an uncountable number of them on the plane? Or is only a countable number?
- 193
5
votes
2
answers
631
views
How do you know when a reflective subcategory of Top is quotient-reflective?
A subcategory $\mathcal{C}$ of the category $Top$ of topological spaces is a reflective subcategory if the inclusion functor $i:\mathcal{C}\hookrightarrow Top$ has a left adjoint $R:Top\rightarrow \...
- 7,193
3
votes
2
answers
908
views
Finite Topology vs sigma Field
Suppose we have a finite $\sigma$ -field $S$, of which $A$ and $B$ are member sets. Since $S$ is closed under union and complementation [by definition], it follows that $(A' \cup B')' = (A \cap B)' \...
- 627
51
votes
5
answers
9k
views
Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 63.1k
24
votes
15
answers
5k
views
Applications of connectedness
In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea of ...
21
votes
1
answer
846
views
Is there a category of topological spaces such that open surjections admit local sections?
The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of $...
- 35.5k
1
vote
2
answers
484
views
Is there good evidence that topological spaces are the correct way to study the general theory of continuity? [closed]
My reason for asking is that the theory of metric spaces is so clean and so many significant theorems can be proved for an arbitrary metric space (which makes it plausible to me that metric spaces are ...
- 4,351
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
1
vote
1
answer
1k
views
Besicovitch Covering Constant for R^1
In the case where $E\subset\mathbb{R}^1$, a Besicovitch cover of $E$ is a cover by open intervals such that each point of $E$ is the center of some interval in the cover.
The Besicovitch Covering ...
- 111
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....