All Questions
5,184 questions
24
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5
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What sets of self-maps are the continuous self-maps under some topology?
An open question on MSE, https://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a ...
- 511
24
votes
9
answers
2k
views
Self-containing structures
This question is partly inspired by this question: independently of the original context, I'm interested in the general claim* that an ill-founded set theory would represent certain mathematical ...
24
votes
1
answer
1k
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What topological principle is at work here?
[I'm cross-posting this from MSE. I initially asked there 10 days ago, and the question was well-received, but left unanswered.]
My question is inspired by a problem I discovered in Putnam and Beyond,...
- 956
24
votes
0
answers
918
views
The topologies for which a presheaf is a sheaf?
Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on $...
- 8,669
24
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0
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751
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Are amenable groups topologizable?
I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
- 3,897
24
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0
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2k
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Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property, without Choice
While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
user5810
23
votes
13
answers
7k
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What should be taught in a 1st course on smooth manifolds?
I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic ...
23
votes
3
answers
9k
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Sets with positive Lebesgue measure boundary
Consider a compact subset $K$ of $R^n$ which is the closure of its interior. Does its boundary $\partial K$ have zero Lebesgue measure ?
I guess it's wrong, because the topological assumption is ...
- 18.7k
23
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12
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18k
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A book in topology
I will have to teach a topology course:
it starts in point set topology and ends at fundamental group of $S^1$.
In the past I have used two different books:
Elementary Topology. Textbook in Problems,...
23
votes
4
answers
1k
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Is $\beta \mathbb{N}$ homeomorphic to its own square?
Let $\mathbb{N}$ be the set of natural numbers and $\beta \mathbb N$ denotes the Stone-Cech compactification of $\mathbb N$.
Is it then true that $\beta \mathbb N\cong \beta \mathbb N \times \beta \...
- 231
23
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3
answers
4k
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Continuous functions taking uncountably many values countably often
Let $f$ be a continuous function defined on the closed interval $[0,1]$. Clearly $f$ is bounded and attains its bounds.
Then my question is how often can $f$ take a value in its range countably many ...
- 4,862
23
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5
answers
2k
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The "right" topological spaces
The following quote is found in the (~1969) book of Saunders MacLane,
"Categories for the working mathematician"
"All told, this suggests that in Top we have been studying
the wrong mathematical ...
- 18.7k
23
votes
4
answers
1k
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Spaces with no topological monoid structure which are homotopy equivalent to topological monoids
In motivating $A_\infty$-spaces to my students I'm going to insist on the homotopy invariance of the notion, saying that "being $A_\infty$ is the homotopy invariant version of being a topological ...
- 6,777
23
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4
answers
2k
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Which is the correct ring of functions for a topological space?
There is a fact that I should have learned a long time ago, but never did; I was reminded that I did not know the answer by Qiaochu's excellent series of posts, the most recent of which is this one.
...
- 54.6k
23
votes
6
answers
2k
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Is there a topological description of combinatorial Euler characteristic?
There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
- 54.6k
23
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1
answer
2k
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Is the normal bundle of a torus trivial?
Question:
Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?
What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $...
- 501
23
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1
answer
1k
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Which spaces have the (weak) homotopy type of compact Hausdorff spaces?
Inspired by the discussion in the comments of this question, I'd like to ask the following question: is it possible to characterize the class of spaces that are homotopy equivalent (or weak equivalent)...
- 31.2k
23
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3
answers
2k
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An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request
There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter.
But there's ...
- 27.7k
23
votes
1
answer
706
views
Which ordered fields are homeomorphic to their power?
It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ...
- 6,741
22
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4
answers
6k
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Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
An anonymous question from the 20-questions seminar:
Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
- 1,059
22
votes
8
answers
3k
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Connections between ultrafilters in topology and logic
I have a some-what vague question. It seems to me that there are two main ways in which ultrafilters (on a set) can be used. One is in topology. The notion of an ultrafilter converging to a point is ...
- 15.5k
22
votes
6
answers
5k
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Topological characterization of the closed interval $[0,1]$
This question is related to question 92206 "What properties make $[0, 1]$ a good candidate for defining fundamental groups?" but is not exactly equivalent in my opinion. It is even suggested in one ...
22
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2
answers
1k
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Is every closed set of Q² the intersection of some connected closed set of R² with Q²
Let $F\subset\mathbb{Q}^2$ a closed set. Does there exists some closed and connected set $G\subset\mathbb{R}^2$ such that $F=G\cap\mathbb{Q}^2$?
For example if $F=\{a,b\}$, you can take $G$ the ...
- 3,033
22
votes
2
answers
977
views
Mapping from a finite index subgroup onto the whole group
Dear All,
here is the question:
Does there exist a finitely generated group $G$ with a proper subgroup $H$ of finite index, and an (onto) homomorphism $\phi:G\to G$ such that $\phi(H)=G$?
My guess ...
- 1,437
22
votes
5
answers
7k
views
Defining a topology in the Power Set
I have the following question:
Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$.
If the ...
- 335
22
votes
3
answers
2k
views
What is a TMF in topology?
What is a topological modular form? How are they related to 'normal' (number-theoretic) modular forms?
- 15.1k
22
votes
2
answers
1k
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Can a continuous real-valued function on a large product space depend on uncountably many coordinates?
Is there a reasonably well-behaved topological space $X$ (ideally Polish), a set $\kappa$, and a continuous function $g: X^\kappa\to\mathbb{R}$ that depends on uncountable many coordinates?
If $X$ is ...
22
votes
2
answers
1k
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Toposes (topoi) as classifying toposes of groupoids
A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Butz and Moerdijk have shown that if the topos has ...
- 38.6k
22
votes
1
answer
754
views
Undetermined Banach-Mazur games in ZF?
This question was previously asked and bountied on MSE, with no response. This MO question is related, but is also unanswered and the comments do not appear to address this question.
Given a ...
- 20.5k
22
votes
1
answer
712
views
Are $\beta \mathbb{Q}$ and $\beta(\beta\mathbb{Q}\setminus\mathbb{Q})$ homeomorphic?
The canonical inclusion $\beta\mathbb{Q}\setminus \mathbb{Q} \hookrightarrow \beta\mathbb{Q}$ is not the Stone-Čech compactification of $\beta\mathbb{Q}\setminus \mathbb{Q}$. Even so, this doesn't ...
- 1,211
22
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0
answers
676
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Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces?
Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \...
- 15.5k
21
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7
answers
1k
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Reference for topological graph theory (research / problem-oriented)
I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
21
votes
4
answers
4k
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Is every locally connected subset of Euclidean space R^n locally path connected ?
This is not actually a question asked by me. But since I do not know the answer, I would love to know if someone here could answer it.
- 1,598
21
votes
2
answers
3k
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Integer matrices with no integer eigenvalues
Let $$A = \begin{pmatrix} 3&1 \\ 0&1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1&0\\ 1&2 \end{pmatrix}$$ I want to show that the only elements of the semigroup generated by $A$ and $B$...
- 1,045
21
votes
2
answers
2k
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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
21
votes
3
answers
610
views
Which partitions of $[0,1]$ are collection of level sets of a real continuous function?
Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...
- 1,014
21
votes
5
answers
1k
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Explanation for E_8's torsion
To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...
- 15.1k
21
votes
1
answer
759
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Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure
Seen $(\Bbb N,+,\cdot)$ as a semiring, is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-...
- 13.6k
21
votes
1
answer
638
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Grothendieck group of the Fibonacci monoid
Let's denote the Fibonacci numbers by $F_0=0,F_1=1,F_{n+2}=F_{n+1}+F_n \; \forall n \ge 0$. According to Zeckendorf's theorem, every positive integer can be represented uniquely as the sum of some (at ...
- 1,543
21
votes
1
answer
939
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Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related?
It is known that there are only 14 reasonable tensor norms in $Ban$. On the other hand it is well known fact for topologists that one can obtain only 14 different sets from a given set applying ...
- 1,697
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
21
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1
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Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?
Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...
- 37.8k
21
votes
1
answer
2k
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Characterization of Fréchet-Urysohn spaces using sequential continuity at a point
A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$.
$$x_n\to a \qquad \Rightarrow \qquad f(...
- 4,713
20
votes
2
answers
1k
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Rugged manifold
It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary.
Is it still true for topological manifold?
Comments:
To proof the smooth ...
- 45k
20
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1
answer
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A function composed with itself produces the identity
Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity ...
- 2,933
20
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3
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2k
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Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology
The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be represented using ultrafilters.
Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ ...
- 571
20
votes
2
answers
545
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$\kappa$-homogeneous topological spaces
Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-homogeneous if
$|X| \geq \kappa$, and
whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$...
- 48.9k
20
votes
2
answers
1k
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An order type $\tau$ equal to its power $\tau^n, n>2$
(This is a re-post of my old unanswered question from Math.SE)
For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types.
Recall that:
$...
- 6,819
20
votes
2
answers
2k
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Is every compact topological ring a profinite ring?
There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
- 2,210
20
votes
3
answers
2k
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Duality between topology and bornology
I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:
Let $X$ be a set and let $\mathcal{P}(...
- 1,813