All Questions
13,925 questions
37
votes
14
answers
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What are interesting families of subsets of a given set?
Motivation
The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$.
Indeed, one defines a topology on $S$ to be a family of subsets ...
- 33.3k
37
votes
3
answers
3k
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Is there a differentiable map surjective from low to high dimension？
Does there exist a map $f:\Bbb R^n \rightarrow \Bbb R^m$, where $n<m$ and $ n,m \in\Bbb N^+$ such that $f$ is surjective and differentiable?
- 491
37
votes
5
answers
7k
views
Example of sequences with different limits for two norms
I was explaining to my students that if there is an inequality between two norms, then there is an inclusion between their spaces of convergent sequences, with matching limits. I then proceeded to ...
- 2,054
37
votes
5
answers
5k
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Locales and Topology.
As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
37
votes
4
answers
4k
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Which differential equations allow for a variational formulation?
Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\...
- 7,583
37
votes
1
answer
1k
views
Does there exist a continuous 2-to-1 function from the sphere to itself?
I am interested in the following question:
Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$?
I suspect the answer is no, but I don't know ...
- 513
37
votes
5
answers
4k
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Reference for the Gelfand duality theorem for commutative von Neumann algebras
The Gelfand duality theorem for commutative von Neumann algebras states that the following three categories are equivalent:
(1) The opposite category of the category of commutative von Neumann ...
- 37.8k
37
votes
2
answers
2k
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Moving one family of commuting self-adjoint operators to another without losing commutativity on the way
This is actually not a question of mine, so I'll be short on motivation and say nothing beyond that if this were true, a few fancy harmonic analysis techniques that a colleague of mine used in proving ...
- 61.9k
36
votes
3
answers
6k
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In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces.
If it is true that:
In a Topological Space, if there exists a loop that cannot ...
- 4,862
36
votes
2
answers
2k
views
Can non-homeomorphic spaces have homeomorphic squares?
I an wondering if there are non-homeomorphic spaces $X$ and $Y$ such that $X^2$ is homeomorphic to $Y^2$.
- 531
36
votes
4
answers
4k
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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
36
votes
3
answers
10k
views
The deep significance of the question of the Mandelbrot set's local connectedness?
I am given to understand that the celebrated open problem (MLC) of the Mandelbrot set's local connectness has broader and deeper significance deeper than some mere curiosity of point-set topology.
...
- 17.6k
36
votes
4
answers
5k
views
Compact open topology on $\mathrm{Homeo}(X)$
Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)...
- 2,751
36
votes
3
answers
2k
views
Does Euclidean space have a compact factor?
Is $\mathbb{R}^n$ homeomorphic to a product $X \times Y$ with $X$ compact and not a point?
Bing's Dogbone space is a quotient of $\mathbb{R}^3$ with fibers points and arcs, and whose product with $\...
- 10.6k
36
votes
2
answers
2k
views
Is there a "universal" connected compact metric space?
Fact 1. The Cantor set $K$ is "universal" among nonempty compact metric spaces in the following sense: given any nonempty compact metric space $X$, there exists a continuous surjection $f\colon K \to ...
- 22.3k
36
votes
2
answers
2k
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A question about connected subsets of $[0,1]^2$
If $S⊂[0,1]^2$ intersects every connected subset of $[0,1]^2$ with a full projection on the $x$-axis, must $S$ have a connected component with a full projection on the $y$-axis?
An equivalent form：
If ...
36
votes
1
answer
3k
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Is there a general theory of "compactification"?
In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
- 63.9k
36
votes
2
answers
3k
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Computing self-intersections with complex analysis
It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...
- 22.8k
35
votes
13
answers
24k
views
Examples of non-metrizable spaces
I want to know some examples of topological spaces which are not metrizable. Of course one can construct a lot of such spaces but what I am looking for really is spaces which are important in other ...
35
votes
2
answers
5k
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Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?
This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...
- 2,974
35
votes
4
answers
6k
views
How are infinite-dimensional manifolds most commonly treated?
I originally posted this question on StackExchange, where it was suggested I post here. It was also suggested I read about Hilbert manifolds and Fréchet manifolds. Nevertheless, I am still looking for ...
- 655
35
votes
2
answers
2k
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Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 156k
35
votes
2
answers
9k
views
tr(ab) = tr(ba)?
It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace ...
- 43.2k
35
votes
1
answer
2k
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Which polygons can be turned inside out by a smooth deformation?
Take a non-degenerate polygon with side lengths $\{a_1,\dots,a_n\}$ in a convex configuration. What is the condition on the $a_i$'s so that the polygon can be turned inside out by a continuous motion ...
- 4,862
35
votes
4
answers
4k
views
An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...
- 6,917
35
votes
1
answer
1k
views
Chromatic number of a topological space
Here is a question I asked myself years ago. Since it is not really in my field, I hope to find some (partial) answers here... Since it was unclear, I precise that I am looking for an answer in ZFC, ...
- 1,005
35
votes
1
answer
2k
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A Topology such that the continuous functions are exactly the polynomials
(I originally asked this question on Math.SE, where it received a lot of attention, but no solution.)
Which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous ...
- 3,017
34
votes
6
answers
4k
views
Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
- 737
34
votes
8
answers
9k
views
When is a Banach space a Hilbert space?
Let $\mathcal{X}$ be a real or complex Banach space.
It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.
...
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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
34
votes
4
answers
3k
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In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent?
I understand that one can give a proof of each of these propositions assuming the truth of the other. But this seems a bit squishy to me, since there is a trivial sense in which any two true theorems ...
- 19.7k
34
votes
2
answers
3k
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"Transitivity" of the Stone-Cech compactification
Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this ...
- 114k
34
votes
2
answers
4k
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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 ...
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34
votes
2
answers
3k
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Can we recover a von Neumann algebra from its predual?
By definition, a von Neumann algebra is a C*‑algebra A
that admits a predual, i.e., a Banach space Z such that
Z* is isomorphic to the underlying Banach space of A.
(We require that isomorphisms in ...
- 37.8k
34
votes
1
answer
2k
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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
933
views
If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?
I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...
- 1,462
34
votes
1
answer
3k
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tr(ab)=tr(ba), part 2.
This is a Banach space version of Andre Henriques' question
Trace Question
for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
- 31.5k
34
votes
1
answer
4k
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Theme of Isbell duality
Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$....
- 63.1k
34
votes
2
answers
2k
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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 ...
- 41.8k
33
votes
6
answers
13k
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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
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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
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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 ...
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33
votes
4
answers
2k
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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
3
answers
3k
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Reference request for translating from Top to C*-alg
Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality — namely, that the categories of ...
- 18.7k
33
votes
4
answers
2k
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Hahn-Banach theorem with convex majorant
At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\...
- 16.4k
33
votes
2
answers
1k
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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
11k
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Counterexample for the Open Mapping Theorem
I would like to ask a counterexample for the classical theorem in functional analysis: the open mapping theorem in the case that $Y$ is Banach, but $X$ is not Banach to show that the completeness of X ...
- 331
33
votes
4
answers
7k
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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
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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
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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, ...
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