Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,601 questions
34
votes
2
answers
4k
views
How do you axiomatize topology via nets?
Let $X$ be a set and let ${\mathcal N}$ be a collection of nets on $X.$
I've been told by several different people that ${\mathcal N}$ is the collection of convergent nets on $X$ with respect to some ...
- 691
31
votes
17
answers
14k
views
Applications of Brouwer's fixed point theorem
I'm presenting Brouwer's fixed point theorem to an audience that knows some point-set topology. Does anyone have any zippy / enlightening / cool applications or consequences of it? So far, I have:
...
5
votes
7
answers
4k
views
Do the empty set AND the entire set really need to be open? [closed]
My question is motivated by the previous discussion 'Why is a topology made of open sets?'. While the axioms for arbitrary unions and finite intersections are without doubt essential to the concept of ...
- 7,127
2
votes
0
answers
223
views
Is the realization of a proper map of simplicial spaces proper ?
Let $f:X \rightarrow Y$ be a map of $m$-dimensional simplicial spaces (which means that all simplices above dimension $m$ are degenerate). Recall, that $f$ is a natural transformation of functors from ...
- 11.1k
333
votes
34
answers
96k
views
Why is a topology made up of 'open' sets? [closed]
I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ...
21
votes
2
answers
2k
views
Colimits in the category of smooth manifolds
In the category of smooth real manifolds, do all small colimits exist? In other words, is this category small-cocomplete? I can see that computing push-outs in the category of topological spaces of ...
- 1,162
2
votes
0
answers
270
views
Homotopy equivalences and cores
Hi all,
Before asking my question, I need to fix some terms and notation.
Let $M$, $M'$ be locally compact, Hausdorff spaces, and $f:M\rightarrow M'$ a homotopy equivalence with homotopy inverse $g:...
- 93
3
votes
3
answers
699
views
Is there a name for this property of a topology?
This property seems like it should have a nice name, but I can't find one anywhere. Does anyone know a name for this?
For each non-empty open set $U$, there exist proper open subsets $\{U_i\}_{i\in ...
- 1,488
3
votes
1
answer
3k
views
Is a proper quotient map closed ?
I am trying to produce closed quotient maps, as they allow a good way of creating saturated open sets (as in this question).
A map $f:X\rightarrow Y$is called proper, iff preimages of compact sets ...
- 11.1k
0
votes
3
answers
248
views
how slow can the dimension of a product set grow?
Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:
$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,
where $\sim$ denotes "...
- 1,839
10
votes
2
answers
3k
views
Continuous function from $[0,1]$ to $[0,1]$
Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
- 1,001
19
votes
3
answers
5k
views
Is a inverse limit of compact spaces again compact ?
Then one can construct a model for the inverse limit by taking all the compatible sequences.
This is a subspace of a product of compact spaces. This product is compact by Tychonoff. If all the spaces ...
- 11.1k
9
votes
1
answer
718
views
What is enough to conclude that something is a CW complex?
This question was something I considered when looking into CW-structures on Grassmannians, but I found no general treatment of this in the literature:
Question: Assume that $X$ is an $n-1$ ...
- 2,590
3
votes
2
answers
601
views
SU(2) representations of alternating knot groups
Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...
- 1,129
9
votes
3
answers
952
views
Is there a non-trivial topological group structure of $\mathbb{Z}$?
More specificaly, is there a haussdorf non-discrete topology on $\mathbb{Z}$ that makes it a topological group with the usual addition operation?
- 1,001
12
votes
2
answers
2k
views
Is this a known compactification of the natural numbers?
Given two infinite sets $A$, $B$ of natural numbers, write $A\preceq B$ if $B\setminus A$ is a finite set. Define the equivalence relation $A\sim B$ if $A\preceq B$ and $B\preceq A$, and let $\partial\...
- 9,366
4
votes
2
answers
3k
views
What do you call a topology that is closed under arbitrary intersections?
An arbitrary union, or a finite intersection, of open sets in a topological space is again open. What name is given to the hypothetical property that an arbitrary intersection of open sets is open?
...
- 9,366
7
votes
2
answers
419
views
Relation between $KO$ and $K$
What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
1
vote
2
answers
686
views
Existence of convergent subsequences for all values in range?
Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in \[-1,1\]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps just for the open ...
- 367
30
votes
5
answers
2k
views
Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?
Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)?
If not true in general, is there any condition ...
6
votes
2
answers
2k
views
References and applications involving the Krull Toplogy
I was wondering if anyone can suggest a reference which treats the Krull topology. Most of the books I have found don't go into any kind of detail.
It is my understanding that the Krull topology ...
- 115
28
votes
5
answers
4k
views
Two-to-one continuous mapping from R² to R²
Hello. I have a question.
Does there exist a continuous mapping
$F:\mathbb{R}^2\rightarrow\mathbb{R}^2$
such that for every $c\in F(\mathbb{R}^2)$
there are two and only two points $z_{1}$, $z_{2}$...
- 301
107
votes
9
answers
36k
views
solving $f(f(x))=g(x)$
This question is of course inspired by the question How to solve f(f(x))=cosx
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
- 41.4k
2
votes
1
answer
1k
views
Finding saturated open sets
Suppose I have a continuous map $f:X\rightarrow Y$. Then one can wonder, whether for every open set $U\subset X$ the set
$U':=\{x\in X|f^{-1}(f(x))\subset U\}$ is open again. This is not true in ...
- 11.1k
10
votes
2
answers
367
views
existence of a connected set with given connected projections.
Suppose A and B are compact connected sets in the XY plane and XZ plane respectively in R^3. Suppose further that the the range of x-values taken by A and B are the same (i.e, projections of A and B ...
- 515
16
votes
5
answers
6k
views
Regular spaces that are not completely regular
In the undergraduate toplogy course we were given examples of spaces that are $T_i$ but not $T_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T_3$ but not $T_{3.5}$ was given. ...
- 2,104
6
votes
4
answers
1k
views
Fundamental domains of measure preserving actions
Suppose a finite group $G$ acts on a standard probability space $(X, \mu)$ by measure-preserving actions (i.e. $\mu(g(A)) = \mu(A)$ for all $g \in G$ and $A \subset X$ measurable). In addition, ...
- 629
4
votes
1
answer
2k
views
Fiber bundle = principal bundle + fiber?
This question is heavily related to this question.
Fix a sufficiently nice and connected topological space $B$ and let $FB$ be the category of fiber bundles over $B$. A morphism $f: (E\to B)\to (E'\...
- 1,085
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
7
votes
3
answers
2k
views
Using topology to characterize embedded Lie subgroups of Lie groups.
Cartan's theorem states that any topologically closed subgroup of a Lie group is an embedded Lie subgroup.
This leads us to ask the following question:
Can we replace "topologically closed" with a ...
- 2,374
33
votes
4
answers
2k
views
Can a connected planar compactum minus a point be totally disconnected?
What the title said. In a slightly more leisurely fashion:-
Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x\}$ be ...
- 25k
31
votes
7
answers
5k
views
Why is it useful to classify the vector bundles of a space?
It seems to me that vector bundles are useful because they allow us to bring to bear all of the linear algebra we know to aid in the study of topological spaces. Now, I've read somewhere that it is ...
- 775
7
votes
2
answers
2k
views
Intersection form in twisted homology (homology with local coefficients)
The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
- 1,129
17
votes
1
answer
2k
views
Which Fréchet manifolds have a smooth partition of unity?
A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is:
Which Fréchet manifolds have a smooth partition of unity?
How is the ...
- 4,519
80
votes
5
answers
6k
views
How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?
This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
...
- 236k
11
votes
5
answers
1k
views
Confusion over a point in basic category theory
"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological ...
- 1,207
30
votes
8
answers
3k
views
Cryptomorphisms
I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are
Topological Spaces. These can be defined in terms of open sets, ...
4
votes
1
answer
417
views
"Category" of Nonempty Metric Spaces and Contractive Maps?
The usual way of getting a category of metric spaces is to take metric spaces as objects, and the nonexpansive maps (ie, functions $f : A \to B$ such that $d_B(f(a), f(a')) \leq d_A(a, a')$) as ...
- 9,201
2
votes
2
answers
1k
views
Simple question of topological cofibration
I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the ...
- 367
32
votes
3
answers
4k
views
Which spaces are inverse limits of discrete spaces ?
There is the following theorem:
"A space $X$ is the inverse limit of a system of discrete finite spaces, if and only if $X$ is totally disconnected, compact and Hausdorff."
A finite discrete space ...
- 11.1k
2
votes
1
answer
336
views
Topologies making a class of functions continuous [closed]
Let $X:=\{f: \mathbb{C}\to \mathbb{C}\}$ be a class of total functions on $\mathbb{C}$ closed under composition, addition, multiplication, and scalar multiplication. Does there exist a topology on $\...
6
votes
3
answers
372
views
Notion of finite dimensional simplicial space
I was wondering, what a $N$-dimensional simplicial space $X$ should be. Of course the degeneracy maps force the spaces to be nonempty in high dimensions. Currently I have two different versions and i ...
- 11.1k
38
votes
7
answers
5k
views
Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)
In this question, Harry Gindi states:
The fact that a commutative ring has a natural topological space associated with it is a really interesting coincidence.
Moreover, in the answers, Pete L. ...
- 21k
7
votes
3
answers
1k
views
The continuous as the limit of the discrete
Reading this documment: www.math.ucla.edu/~tao/preprints/compactness.pdf, I got interested in the following thing: "One can also use compactifications to view the continuous as the limit of the ...
- 153
52
votes
7
answers
8k
views
"Algebraic" topologies like the Zariski topology?
The fact that a commutative ring has a natural topological space associated with it is still a really interesting coincidence. The entire subject of Algebraic geometry is based on this simple fact.
...
-3
votes
2
answers
1k
views
Finite versus infinite on non-Hausdorff topologies [closed]
Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...
- 283
13
votes
2
answers
646
views
Functions separting points in Hausdorff spaces
A colleague in algebra asked me this, and I couldn't answer it. On the Wikipedia page for "epimorphism" it is claimed that in the category of Hausdorff spaces and continuous maps, a function is epi ...
- 18.7k
5
votes
1
answer
723
views
Sheaf condition and representability in the category Top
This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{...
- 7,442
3
votes
6
answers
3k
views
Cone in a metric space
We know the definition of a cone in a Real Banach Space.
I want to know if there is any definition for a cone in an abstract metric space.
Have you ever seen such definition anywhere?
- 520
7
votes
1
answer
3k
views
definition of the end of a manifold?
I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function assigning to each compact set K a conected component e(K) of the ...
- 71