All Questions
13,927 questions
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Localic locales? Towards very pointless spaces by iterated internalization.
One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points. Of course when one studies locales, one "actually" studies frames,
certain sorts of ...
- 17.6k
14
votes
4
answers
1k
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Obtain any 3-manifold from repeating surgeries on knots in $S^3$
In Witten's “QFT and Jones Polynomials” paper, page 383, it states that: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) ...
- 755
14
votes
5
answers
4k
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Is the long line paracompact?
A manifold is usually defined as a second-countable hausdorff topological space which is locally homeomorphic to Rn. My understanding is that the reason "second-countable" is part of the definition is ...
- 24.1k
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2
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1k
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Hausdorff spaces with trivial automorphism group
Is the singleton space the only Hausdorff space $X$ such that the set of automorphisms $\varphi: X\to X$ equals $\{\textrm{id}_X\}$?
- 48.9k
14
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3
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1k
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Quotient of solid torus by swapping coordinates on boundary
Let $T$ be the solid 2-torus and let $\sim$ be the equivalence relation on $T$ generated by the relation $\{(\alpha,\beta) \sim (\beta,\alpha) \mid \alpha, \beta \in S^1\}$ on the boundary $\partial T=...
- 1,813
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6
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6k
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Russian Equivalent of Big Rudin
Is there any Russian-authored textbook on Analysis equivalent to Big Rudin (Real and Complex Analysis)?
I like Russian math textbooks a lot. I am looking for Russian textbooks (either in English or ...
- 149
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3
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412
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$T_2$ topologies that are "as disjoint as possible"
Let $X$ be an infinite set. Are there Hausdorff topologies $\tau_1, \tau_2$ on $X$ such that $\tau_1\cap\tau_2 = \{\emptyset\} \cup \{U\subseteq X: X\setminus U\text{ is finite}\}$?
(That is, the ...
- 48.9k
14
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4
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2k
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Products of Baire spaces
I could not find any references about this fact. I apologize if this is completely trivial, but is the product of two Baire spaces, or for that matter of finitely many of them a Baire space? Now is a ...
- 3,804
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6
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3k
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What's a natural candidate for an analytic function that interpolates the tower function?
I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that ...
- 4,466
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4
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550
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About the existence of characters on $B(X)$
Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$?
I know the proof of the fact that $M_n(\mathbb{C})$ ...
- 355
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2
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2k
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Well-pointed space which is not locally contractible
I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...
- 6,210
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5
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4k
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Is there an extension of the Arzela-Ascoli theorem to spaces of discontinuous functions?
The Arzela-Ascoli function basically says that a set of real-valued continuous functions on a compact domain is precompact under the uniform norm if and only if the family is pointwise bounded and ...
- 943
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3
answers
1k
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What is a monoidal metric space?
At time of writing, the highest rated answer to my question What is a metric space? is Tom Leinster's account of Lawvere's description of a metric space as an enriched category. This prompted my ...
- 26.8k
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2
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1k
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Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$?
Question: Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$? I'd guess that the answer doesn't depend on choice of differentiable structure, but maybe it does.
Motivation:...
- 787
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2
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761
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Is there a large colimit-sketch for topological spaces?
Question. Is there a large colimit-sketch $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$?
In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}...
- 63.1k
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2
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892
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Do distance functionals separate probability measures?
Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance ...
- 459
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1
answer
1k
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Minimal good cover of the torus
Recall that an open cover $\mathfrak{U} = \{ U_\alpha \}$ of a manifold $M$ is called a good cover if all possible finite intersections $U_{\alpha_1} \cap ... \cap U_{\alpha_n}$ are contractible.
...
- 253
14
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2
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6k
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Are weak and strong convergence of sequences not equivalent?
For some infinite-dimensional Banach spaces $E$, it is easy to find sequences $\langle x_i:i\in\mathbb N_0\rangle$ which converge to zero weakly but not in the norm topology, i.e. we have $\lim_{i\to\...
- 3,584
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3
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What spaces can be obtained from $\mathbb{R}^{n}$ by taking quotient spaces and subspaces?
Is there a good characterization of the smallest collection of topological spaces which contains $\mathbb{R}^{n}$ for each $n$, and is closed under taking subspaces and quotient spaces?
A bit of ...
- 726
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2
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876
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Is there a natural topology for sets of topological spaces?
The Gromov–Hausdorff metric makes a set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is ...
- 691
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2
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1k
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Are smooth functions tame?
I know the article of Hamilton on the inverse function theorem of Nash and Moser (with the same title) where he proves that $C^\infty(M)$ is a tame Fréchet space, when $M$ is closed or compact with ...
- 9,853
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4
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3k
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Representing a product of matrix exponentials as the exponential of a sum
In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices ...
- 28.6k
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2
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2k
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Is the composition of two nowhere differentiable functions still nowhere differentiable?
Let $f,g:\mathbb R\to\mathbb R$ be two continuous but nowhere differentiable functions. By the Denjoy–Young–Saks theorem for almost every point $x_0\in\mathbb R$ one has
$$
\limsup\limits_{x\to x_0}\...
- 938
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1
answer
459
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A parametric version of the Borsuk Ulam theorem
Is there a topological space $X$, which is not a singleton, and satisfies the following property?
For every continuous function $f: X\times S^2\to\mathbb{R}^2$ there exist a point $x\in S^2$ such ...
- 356
14
votes
3
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547
views
Recognizing Lipschitz functions up to change of target metric
Let $K$ be a compact subset of $\mathbb{R}^n$ (for simplicity, I am happy to take $K=\overline{B(0,1)}$ for now if it is easier).
Let $f:K \rightarrow \mathbb{R}^m$ be a continuous function.
Is ...
- 143
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2
answers
502
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Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?
Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder.
The map $j:n\...
- 63.9k
14
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2
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1k
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Baire Category Theorem for complete uniform spaces
The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
- 1,270
14
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3
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6k
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What is the definition of continuity of set-valued functions?
According to the wiki of Kakutani's fixed-point theorem, A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W \...
- 143
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3
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3k
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The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$
Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
- 1,373
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1
answer
578
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Obstruction of spin-c structure and the generalized Wu manifods
Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the
$$
H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
- 2,147
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1
answer
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Stone-Weierstrass Theorem without AC
To what extent does the usual Stone-Weierstrass Theorem depend on some form of the Axiom of Choice? There seems to be a lot of literature on constructive versions in toposes, but I have been unable ...
- 2,049
14
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1
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581
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How “disconnected” can a continuum be?
A continuum is a compact connected metrizable topological space.
Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint ...
- 3,011
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2
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974
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Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?
I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
14
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2
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490
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Which spaces have enough curves
Let $\mathbf{Top}$ be the category of topological spaces, and let $I\in\mathbf{Top}$ be the unit interval $I=[0,1]\subset\mathbb{R}$. For any space $X$, let $|X|$ denote the underlying set of points; ...
- 8,669
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1
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Distributivity of group topologies on $\Bbb Z$
Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1].
Is $(\mathcal L,\subseteq)$ distributive?
$$~$$
[1] ...
- 1,145
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2
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723
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Why do the projections in the Calkin algebra not form a lattice?
Let $H$ be an infinite dimensional separable complex Hilbert space. Denote by $\mathcal{B}(H)$ the C*-algebra of bounded operators on $H$, $\mathcal{K}(H)$ the ideal of compact operators on $H$, and $\...
- 3,115
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votes
2
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4k
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Direct construction of the Stone-Čech compactification using ultrafilters?
If $X$ is a set (regarded as a discrete space), its Stone-Čech compactification can be identified with the set of ultrafilters on $X$ with its natural (Stone) topology. If $X$ is a general ...
- 118k
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830
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Spectrum of matrix involving quantum harmonic oscillator
The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator
$$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$
Fix two numbers $\alpha,\beta \...
- 192
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1
answer
694
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Criterion for a Banach algebra to be finite dimensional
Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional.
Question. Does it follow that $A$ is finite dimensional?
...
- 12.6k
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1
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604
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Continuum Hypothesis and the fact that every co-finite topological space, with uncountable underlying set , is contractible
Let $X$ be a co-finite topological space. If $|X| \ge 2^{\aleph_0}=\mathfrak c$, then $X$ is contractible (https://en.wikipedia.org/wiki/Contractible_space) . Indeed, there is a bijection $f: X \times ...
user111524
14
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4
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1k
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Is every continuous microlocal operator a pseudo-differential operator?
Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space.
Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal.
By being microlocal I mean that the wave ...
- 8,086
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1
answer
2k
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Infinite tensor product of states
Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher.
Even in the simplest ...
- 143
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237
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Must a path of compact sets in $X$ descend to a path in $X$?
(I am most interested in the case $X=\mathbb R^2$, but of course one could ask the same question for manifolds, or metric spaces in general.)
Let $\text{Com}(\mathbb R^2)$ denote the space of nonempty ...
- 721
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4
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$L^p$ norm means
Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed ...
- 96.4k
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1
answer
685
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Quotient of Three Dimensional Torus by Permutation on Coordinates
The Mobius Strip can be realized as a quotient of $T = (S^1)^2$ via the identifications $(x,y) \sim (y,x)$.
I tried to generalized this concept to a higher dimension, and consider the quotient of $(...
- 943
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votes
2
answers
2k
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Is this property equivalent to Lusin's property (N) for continuous functions?
A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies Lusin's (N) property if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is ...
- 1,601
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2
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926
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"Explicit" embedding of $\ell^1$ as a closed subalgebra of a direct sum of matrix algebras
For sake of brevity let $A$ denote the Banach algebra formed by equipping $\ell^1({\mathbb N})$ with pointwise multiplication. This algebra is clearly not isomorphic as a Banach algebra to any uniform ...
- 25.8k
14
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1
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571
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"Scott completion" of dcpo
If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U $ for ...
- 42.4k
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2
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843
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Proper topological spaces
Recall that a topological space is ccc, or has the countable chain condition, if every family of pairwise disjoint open sets is countable.
But equivalently, we can say that the forcing defined with ...
- 39.8k
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2
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413
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Given a sequence of reals, we can find a dense sequence avoiding it, but can we find one continuously?
Let $S$ be the set of injective sequences in $\mathbb{R}$:
$$S = \{s: \mathbb{N} \rightarrow \mathbb{R}: s(m) \neq s(n) \text{ if }m \neq n\}.$$
Consider $S$ with the topology of pointwise convergence,...
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