All Questions
1,339 questions with no upvoted or accepted answers
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138
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Does any smooth oriented closed orbifold have a fundamental class
This thread:triangulation of orbifolds
has shown that any smooth closed orbifold has a triangulation. My further question is: if the difference of any two triangulations $P$ and $Q$ is a boundary of a ...
- 781
2
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0
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252
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Special orthogonal groups over spheres
In Norman Steenrod's book "The Topology of Fibre Bundles", on page 37, one can find the following conjecture: if $n$ is a power of two then the fibre bundle with the projection $SO(n)\to SO(n)/SO(n-1)=...
2
votes
0
answers
114
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Linear topologies on the finite dual of the polynomial algebra
Let $\Bbbk[X]$ be the polynomial algebra in one indeterminate over a field $\Bbbk$, endowed with the primitive-like bialgebra structure, i.e. $\Delta(X)=X\otimes1+1\otimes X$ and $\varepsilon(X)=0$.
...
- 718
2
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80
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Does compact Hausdorff exponentiable topology on opens of $X$ induce a compact Hausdorf topology on $C(X,Y)$
My question arose after I read Topologies on spaces of continuous functions of Martin Escardo and Reinhold Heckmann. Terminology of the question is similar to note. I avoid here of definitions of all ...
- 774
2
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171
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Existence of compatible almost complex structure of symplectic fibration with nice property
Let $\pi: E \to B$ be a fibration over a closed surface $B$ with fier $g(F) \ge 2$. Suppose that both of $B$ and $F$ are closed surfaces and $g(B) \ge2$ and $g(F) \ge 2$. Fix a Kahler structure $(...
- 185
2
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51
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Existence of a canonical embedding from a quotient of $\mathscr{F}^\ast(\mathcal A(H))$ to another
Let $H$ be a multiplicative, commutative monoid, and denote by $H^\times$ the group of units of $H$, by $\mathcal A(H)$ the set of atoms (or irreducible elements) of $H$, by $\mathscr{F}^\ast(\mathcal ...
- 10.5k
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305
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Are homotopy equivalent manifolds with homeomorpic boundaries themselves homeomorphic?
Let $f:M \to M′$ be a homotopy equivalence of topological manifolds with boundary such that $dim(M)=dim(M′)$ and $f:\partial M \to \partial M′$ is a homeomorphism. Does this imply the existence of a ...
- 79
2
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82
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Uniquely divisible neighborhoods of identity in topological groups
Let $G$ be a (finite dimensional real) Lie group, and let $A\subset G$ be an open neighborhood of identity. If $A=\operatorname{Exp}(\mathcal{A})$ is the injective range of the exponential map from a ...
- 1,959
2
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603
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Valuation topology vs modified valuation topology
Let $K$ be a field with valuation $v:K\to G\cup\{\infty\}$ where $G$ is an ordered abelian group. In section 7.62 of the book "Foundations of analysis over surreal number fields." Vol. 141. Elsevier, ...
- 596
2
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83
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Sheaf of R-modules and modules over compactly supported functions
I'm looking for a reference for the following result:
Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over ...
- 42.4k
2
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106
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Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
- 6,700
2
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109
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Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from two different diffeomorphisms
We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end).
Given an embedded torus $T$ with trivial normal ...
- 10.5k
2
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279
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Can a bounded open set in $R^n$ be always approximated from outside with a finite union of dyadic cubes?
Suppose we have a bounded open set $S$ in $R^n$. Consider the collection of closed dyadic cubes $C_k$'s (https://en.wikipedia.org/wiki/Dyadic_cubes). I was wondering if there always exists a finite ...
- 131
2
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331
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Why is a component of complement of Jordan curve 1-connected w/o Schoenflies?
The complement of a simple closed curve in the Riemann sphere has two connected components (Jordan). Schoenflies theorem implies that each of these components is homeomorphic to a disk and hence each ...
- 494
2
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73
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Dual equivalence for multioperators
This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...
- 1,017
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68
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Topologies with the same convex closed sets
Let $\tau_1$ and $\tau_2$ be locally convex Hausdorff topologies on vector space $X$ such that $(X,\tau_1)^\ast = (X,\tau_2)^\ast$. It is well known that $(X,\tau_1)$ and $(X,\tau_2)$ have the same ...
- 105
2
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178
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Monoid prime ideals and prime congruences
I was wondering what the connection is between the notion of "prime congruence" on a monoid, and the notion of "prime ideal" in a monoid. Starting from a prime ideal $P$ in a monoid $M$, one can ...
- 4,547
2
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146
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Existence of enough local sections
Let $\pi: G\to X$ be a continuous open (!) surjection of locally compact Hausdorff spaces. Assume that each fiber $G_x=\pi^{-1}(x)$, $x\in X$ carries a group structure making it a locally compact ...
user1688
2
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160
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Pre-cosheaf of connected components
Consider a continous map $f:Y \to X$ between topological spaces. The pre-cosheaf $\mathcal{F}: Open(X) \to Set$ of connected components of the inverse image is defined as $\mathcal{F}(U):= \pi_0(f^{-1}...
- 229
2
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208
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A functor on the category of rings, algebras or compact Hausdorff topological space
Assume that $R$ is a unital ring or a complex or real (Banach or $C^{*}$) algebra.
We define a relation $M$ on $R$ as follows: $$a\;M b \;\;\; \text{iff}\;\; a=xy,\;b=yx \;\; \text{for ...
- 356
2
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139
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Centralizer of a dense subgroup in a maximal subgroup of a reductive group
I am looking for a reference to the following statement
"Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...
- 21
2
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360
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Given locally compact and $\sigma$-compact, can we get partition of unity?
Let $X$ be a locally compact, $\sigma$-compact Polish (complete and separable metric) space. How to prove: "There is an increasing sequence of continuous cut-off functions with compact support, $0\...
- 471
2
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answers
439
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Quotients of simplicial complexes which are simplicial complexes
In the category of topological spaces, I would like to know that quotients of simplicial complexes (or $\Delta$-complexes) by equivalence relations which are "unramified" in a suitable sense still ...
- 31
2
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224
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cross-sections of a sphere bundle
Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...
- 2,223
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467
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Reference request: The compactness and compact embedding in Besov Space?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...
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119
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Are all locally compact anisotropic groupoids etale up to equivalence?
By groupoid I mean "open topological groupoid",i.e. topological groupoids whose source and target maps are open surjections, and the notion of equivalence I'm considering is the isomorphism in the ...
- 42.4k
2
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87
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Terminology for torsion semigroups where the order of elements is uniformly finite
A (multiplicatively written) semigroup $\mathbb A = (A, \cdot)$ with the property that ${\rm ord}_\mathbb{A}(a) := |\{a^n: n \in \mathbf N^+\}| < \infty$ for every $a \in A$ is called a periodic (...
- 10.5k
2
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answers
355
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Existence of topology on the space of continuous functions
Let $C:=C([0,1],\mathbb{R})$ be the space of real-valued continuous functions defined on $[0,1]$. Could we find a topological vector space topology $\pi$ on $C$ such that the following two conditions ...
- 1,835
2
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188
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Space of continuous real-valued functions on $[0,1]^\omega$ with the weak and pointwise topology
Let $C([0,1]^\omega)$ denote the set of continuous functions $f:[0,1]^\omega \to \mathbb{R}$. We endow $C([0,1]^\omega)$ with two topologies. Let $\tau$ be the pointwise topology on $C([0,1]^\omega)$ ...
- 48.9k
2
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261
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Normed space that is sigma-totally-bounded but is not sigma-compact
Q1: Is there a separable normed space that is not sigma-compact, but is a countable union of
totally bounded closed subsets?
A test case is the space $C^1(I)$ with the $C^0$ norm where $I=[0,1]$. ...
- 29.1k
2
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183
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Monadicity of profinite algebras
We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples.
In case that I was ...
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2
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329
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Two questions on hyperspace of a metric space
Let $(X,T)$ be a compact metrizable space. For every metric $d$ on $X$ which is $T-$ compatible, the Hausdorff metric on $2^{X}$ gives a topology on $2^{X}$.
(Up to homeomorphism) is this topology ...
- 356
2
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0
answers
99
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Equicontinuity of $\{f_{2n}\circ f_{2n-1}\}$
Let $(X,D)$ be a compact metric space and $\{f_n\}_{n\in\mathbb{N}}$ be a sequence of homeomorphisms of $(X,d)$. It is easy to see that if $\{f_n\}$ is uniformly convergent then $\{g_n\}$ defined by $...
2
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0
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203
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Profinite Topology
Let $V$ and $W$ be pseudovarieties of finite groups. For a finite inverse monoid $M$, the $V$-kernel of $M$ is defined to be the intersection of all sets $f^{−1}(1)$, $f$ is a relational morphism ...
- 421
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0
answers
180
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Pro-p topology on free group
Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...
- 421
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96
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On compactness in $C(X)$
Let $X$ be a Tychonoff space. It is well known, that for a family of scalar functions equicontinuity + pointwise boundedness imply relative compactness in $C(X)$ (with compact-open topology). It is ...
- 5,529
2
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459
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Weak topology on subsets of a normed space
I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset.
When is the norm a continuous function on $E$?
When is the metric induced by the ...
- 5,529
2
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139
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Goldie's Theorem for Semigroups
Goldie's theorem is a theorem in noncommutative ring theory that gives a clear picture of semiprime Noetherian rings (actually a slightly broader class). Let $R$ be a semiprime Noetherian ring. The ...
- 6,870
2
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87
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Local section of Lie Groupoids
Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...
- 21
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516
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The Hausdorff quotient of totally disconnected space
Let $G$ be a second countable locally compact Hausdorff group and $X$ be a second countable locally compact almost-Hausdorff $G$-space. If $X$ is totally disconnected and the orbit space X/G is ...
- 1,702
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169
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Classify spaces that make extension theorems hold
Recall a Polish space is a completely metrizable separable space.
Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous ...
- 6,287
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104
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Selecting dense diagonals in $\Bbb T^2$
Let $p$ be a prime number and let $G=\bigcup_{n\in \Bbb N}\{\exp(k\frac{2\pi i}{p^n})\mid k\in \Bbb Z\}$ be a Prüfer group. For homomorphisms $f,g:G\to G$ let $H_{f,g}=\{(f(x),g(x))\mid x\in G\}$. ...
- 1,145
2
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343
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continuity with respect to weak-${\ast}$ topology
Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
- 1,835
2
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246
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A possible generalization of the Borsuk Ulam theorem via action of symmetric groups
The symmetric group $S_{m}$ is equiped with the counting Har measure $\mu$ and the obvious sgn character. Assume that $S_{m}$ acts on $S^{n}$, $n\geq m-1$ and $f:S^{n}\to \mathbb{R}^{n}$ ...
- 356
2
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76
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question about a genralized Skorokhod topology
Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$
$$\rho(f,g):=\inf_{\lambda\in\Lambda}\Big\{\max\...
- 1,835
2
votes
0
answers
61
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Removable sets for simply connectedness of a differentiable manifold
I am sorry that my question might be stupid for experts, but I really do not know the answer.
Let $M$ be a smooth $n$-manifold. We assume that there exists a distance $d$ on $M$ such that $(M,d)$ is ...
- 1,881
2
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0
answers
96
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Branch point and alexandrov embeddedness
This is a question I have asked on mathstackexchange with a bounty but without any answer; it is probably more adapted to mathoverflow:
Let us assume that $\Sigma_n$ is a sequence of topological ...
- 914
2
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0
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216
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Standard name for a Monoid/Semigroup with $a+b \leq a, b$?
I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice?
For instance, for reals $a,b > 0$, define $$a \oplus b = \frac{1}{\frac{1}{a}...
- 121
2
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0
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239
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Ostaszewski space's construction Lemma
I'm studying the Ostaszewski's article "On Countably Compact, Perfectly Normal Spaces". I'll add some context. Lemma 1.2 says the following:
Let $X$ be a locally compact, zero-dimensional and ...
user46747
2
votes
0
answers
72
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d-refining covering of normal space
If $X$ is normal, it is well known that for any open-covering $(U_i)$ of $X$, there exist closed subspaces $F_i$ and $G_i$ and an open subspaces $O_i$ such that $$F_i\subset O_i\subset G_i\subset U_i\...
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