All Questions
5,184 questions
34
votes
1
answer
2k
views
Square roots of $\mathbb R^{2n}$
Recently, Richard Dore asked us if $\mathbb R^3$ is the cartesian square of some space, and Tyler Lawson answered beautifully in the negative.
The even powers of $\mathbb R$ were left out in that ...
- 47.3k
34
votes
2
answers
2k
views
Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?
This question was motivated by an answer to this question of Dominic van der Zypen.
It relates to the following classical theorem of Sierpiński.
Theorem (Sierpiński, 1921). For any countable partition ...
- 41.9k
33
votes
6
answers
13k
views
Is a topology determined by its convergent sequences?
Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the ...
- 543
33
votes
4
answers
2k
views
Connectedness in the language of path-connectedness
Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?
A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a ...
- 48.9k
33
votes
6
answers
2k
views
Is there a topology on growth rates of functions?
I've often idly wondered one can say about the collection of "growth rates". By growth rate, let's say we mean an equivalence class of functions $(0,\infty) \to (0,\infty)$, where two ...
- 793
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
33
votes
2
answers
1k
views
can another topology be given to $\mathbb R$ so it has the same continuous maps $\mathbb R\rightarrow \mathbb R$?
We say two topologies $\tau$ and $\rho$ on $X$ are similar if the set of continuous functions $f:(X,\tau) \rightarrow (X,\tau)$ is the same as the set of continuous functions $f:(X,\rho)\rightarrow (X,...
- 1,835
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
33
votes
1
answer
1k
views
Is $L^2(\mathbb R)$ homeomorphic to $L^1(\mathbb R)$?
Is $L^2(\mathbb R)$ homeomorphic to $L^1(\mathbb R)$?
More generally, are there instances of surprising homeomorphisms between non-isomorphic Banach spaces?
- 43.2k
33
votes
2
answers
2k
views
What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
- 30.3k
33
votes
7
answers
4k
views
Topology on the set of analytic functions
Let $H(D)$ be the set of all analytic functions in a region $D$ in $C$ or in $C^n$.
Everyone who worked with this set knows that there is only one reasonable topology
on it: the uniform convergence on ...
- 91.8k
33
votes
2
answers
1k
views
Can $[0,1]^4$ be partitioned into copies of $(0,1)^3$?
Is there a partition of $[0,1]^4$ such that every member of the partition is homeomorphic to $(0,1)^3$?
More generally, I would like to know for which values of $m$ and $n$ there is a partition of $[0,...
- 18.6k
33
votes
1
answer
2k
views
Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?
On page 205 of his Topology textbook, James Munkres made an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...
- 4,597
33
votes
1
answer
3k
views
Fake versus Exotic
Without recourse to the Disc Theorem (or its progeny), is it true that all known examples of exotic differentiable structures on 4-manifolds would be fake rather than exotic?
Terminology (perhaps non-...
- 2,337
33
votes
0
answers
2k
views
Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG?
For a (discrete) monoid $M$, the classifying space $BM$ is the
geometric realization of the nerve of the one object category whose
hom-set is $M$. (This definition gives the usual classfiying space
...
- 5,309
32
votes
4
answers
5k
views
Does the Brouwer fixed point theorem admit a constructive proof?
Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always ...
- 18.7k
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
32
votes
2
answers
5k
views
A question about "Zariski dense" arguments
This question is a little basic, but I think it is consistent with the goals of MO.
My question is about a certain type of argument in algebraic geometry which exploits the abundance of dense sets ...
- 29.2k
32
votes
1
answer
2k
views
Homeomorphisms and disjoint unions
Let $X$ and $Y$ be compact subsets of $\mathbb{R}^n$. Assume that $X \sqcup X \cong Y \sqcup Y$ (here $X \sqcup X$ is the disjoint union of two copies of $X$, considered as a topological space, and ...
- 556
32
votes
1
answer
2k
views
A group allowing exactly 7 group topologies
Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{\text{trivial}}, \mathcal T_{\text{discrete}}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$...
- 1,145
32
votes
5
answers
9k
views
How many binary operations are associative?
Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be ...
- 151k
32
votes
1
answer
2k
views
Bidi: A new cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...
- 3,104
32
votes
1
answer
1k
views
If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?
I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get ...
- 439
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
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:
...
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
31
votes
3
answers
1k
views
Non embedding of $Y\times Y$ into $\mathbb{R}^3$
I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type.
Theorem. Let $Y$ be the union of two segments ...
- 28k
31
votes
6
answers
6k
views
Least number of charts to describe a given manifold
Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it.
E.g. a circle requires at least two charts, and ...
- 2,901
31
votes
4
answers
5k
views
Are all Hawaiian Earrings homeomorphic?
The Hawaiian Earring is usually constructed as the union of circles of radius 1/n centered at (0,1/n): $\bigcup_1^\infty \left[ (0, \frac{1}{n}) + \frac{1}{n}S^1 \right]$. However, nothing stops us ...
- 22.8k
31
votes
13
answers
6k
views
Classic applications of Baire category theorem
I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
31
votes
2
answers
3k
views
Is a normed space which is homeomorphic to a Banach space complete?
I have a normed space $(E,||\cdot||)$ which is homeomorphic (as a topological space) to a Banach space $F$.
Does this imply that $(E,||\cdot||)$ is also a Banach space?
I think I read something ...
- 495
31
votes
2
answers
1k
views
Is $\mathbb{R}\cong\text{Cont}(X,Y)$ for some non-trivial spaces $X,Y$?
For topological spaces $X,Y$ let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y.$ We endow $\text{Cont}(X,Y)$ with the topology inherited from the product topology on $Y^X.$
...
- 48.9k
30
votes
4
answers
2k
views
is f a polynomial provided that it is "partially" smooth?
Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...
- 407
30
votes
5
answers
4k
views
The role of ANR in modern topology
Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...
30
votes
2
answers
2k
views
Does there exist any non-contractible manifold with fixed point property?
Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the ...
- 3,751
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, ...
30
votes
5
answers
3k
views
The ants-on-a-ball problem
Suppose I put an ant in a tiny racecar on every face of a soccer ball. Each ant then drives around the edges of her face counterclockwise. The goal is to prove that two of the ants will eventually ...
- 24.1k
30
votes
2
answers
2k
views
Is every connected subgroup of a Euclidean space closed?
The question listed above (in the context of the complex numbers, but it is a reasonable question to ask in any dimension) was asked by a student in my complex analysis class, and I did not have an ...
- 114k
30
votes
3
answers
2k
views
Is there a subset of the plane that meets every line in two open intervals?
Using the Axiom of Choice, it is possible to construct a subset of the plane that meets every line in two points (these are called "$2$-point sets"). What if, instead of points, we ask for two open ...
- 18.6k
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 ...
29
votes
4
answers
1k
views
Does $M^o=N^o$ imply that $\partial M = \partial N$?
let $M$ be a smooth $n$-manifold with boundary $\partial M$; I denote by $M^o$ the internal part of $M$, that is $M \smallsetminus \partial M$.
The question is the same as in the title: let $M$ and $N$...
- 674
29
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,194
29
votes
1
answer
2k
views
Is there a topological space X homeomorphic to the space of continuous functions from X to [0, 1]?
In general, we might ask when we can find interesting spaces $X, Y$ such that $X$ is homeomorphic to $[X, Y]$. By the Lawvere fixed point theorem $Y$ must have the fixed point property. Happily, $Y = [...
- 118k
29
votes
1
answer
1k
views
Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
- 41.9k
29
votes
1
answer
812
views
Running most of the time in a connected set
Let $P$ be a compact connected set in the plane and $x,y\in P$.
Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small?
...
- 45k
28
votes
8
answers
4k
views
Is there a compact group of countably infinite cardinality?
Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.
Is there a compact (Hausdorff, or even T1) (topological) ...
- 12.6k
28
votes
6
answers
9k
views
Why the triangle inequality?
[Maybe this is asking to be closed; but I thought I'd risk it.]
A metric satisfies the axioms:
$d(x,y)=0$ if and only if $x=y$.
$d(x,y) = d(y,x)$.
$d(x,y) \leq d(x,z) + d(z,y)$.
Similarly (and ...
28
votes
7
answers
13k
views
Regular borel measures on metric spaces
When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
- 18.7k
28
votes
2
answers
2k
views
A property of the unit circle
Let $(X,d)$ be a compact connected metric space with the property that for any distinct points $a,b$, $X\backslash \lbrace a,b\rbrace$ is disconnected. Clearly the unit circle has this property. Is ...
user38122
28
votes
2
answers
5k
views
Is Furstenberg's topology useful?
It's hard not to be amused and perhaps even amazed when first encountering Furstenberg's clever "topological" proof that there are infinitely many primes. Closer inspection, however, reveals ...