All Questions
5,184 questions
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Is there a natural topology for subsets of a fixed topological space?
This question is an extension/clarification of the question: Is there a natural topology for sets of topological spaces?
The Hausdorff distance assigns a distance to any two subspaces $X, Y$ of a ...
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13
votes
2
answers
656
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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 ...
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votes
1
answer
65
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Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
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0
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89
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Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
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249
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+300
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 501
13
votes
1
answer
859
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Mistake on article about Bohr compactification?
$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
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1
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284
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Codimension zero embeddings and maps with small fibers
Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here.
...
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3
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0
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52
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Does there exist a multi-valued "monotone" and "compact" map from a Boolean algebra to the "free" part of $\mathcal{P}(\kappa)$?
This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested:
Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...
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12
votes
1
answer
395
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Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\int_S ...
- 24.2k
7
votes
1
answer
296
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Lower bound on dimension required to disconnect manifold?
This question seems quite classical, but I don't quite know what subarea of topology it falls into.
Suppose that removing the set $S$ disconnects the 2-torus $\mathbb{T}^2 = \mathbb{R}^2\diagup\mathbb{...
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3
votes
1
answer
343
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Fundamental group of the grid on $\mathbb{R}^\mathbb{N}$
The grid on $\mathbb{R}^2$ is defined by the set of points such that at most one coordinate is not an integer. With this in mind, e endow $\mathbb{R}^\mathbb{N}$ with the product topology, where $\...
- 48.9k
10
votes
1
answer
662
views
Are there any tests for knowing whether a topological space admits a CW structure?
We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
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1
answer
144
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Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?
Write $g$ as the inverse of $f$.
Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that
$$
\displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0
$$ as $r\to0$?
If you like,...
- 211
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1
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295
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Connected open sets in the topology generated by the collection of connected open sets
Let $(X,\mathcal{T})$ be a connected topological space. Let $\mathcal{T}'$ be the topology on $X$ that is generated by the collection of connected open sets in $(X,\mathcal{T})$. That is, the ...
2
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157
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About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev
According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result:
Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a
family of ...
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14
votes
1
answer
500
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Is there an 'unnatural' topological construction of an algebraically closed field of positive characteristic?
It's well known that while there is a natural topological construction of a nearly algebraically closed field of characteristic $0$, algebraically closed fields of positive characteristic seemingly ...
- 12.5k
2
votes
1
answer
120
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"Bad" valid edge contractions
In this paper, an edge contraction of a simplicial complex $\Gamma$ is defined as the operation of removing the neighborhood $N_e\Gamma$ of the edge $e=\{0,1\}$ and identifying $N_0\partial N_e\Gamma$ ...
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1
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437
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Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
3
votes
1
answer
132
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Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 48.9k
3
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2
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343
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Cohomology version of Moore space
I asked this question on MSE a few days back but could not get any helpful response. So I am rewriting the post.
It is known to me that given a simply connected finite dimensional (which is also level-...
- 1,177
5
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1
answer
96
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Preimage of a sublocale by a morphism of locales: description by nucleus?
For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs ...
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11
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0
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427
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Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
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3
votes
1
answer
90
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Even covers and collectionwise normal spaces
We call $X$ strongly collectionwise normal if the set $\mathcal{U}_\Delta$ of all neighbourhoods of the diagonal $\Delta_X$ of $X\times X$ is a uniformity. This is equivalent to the property that for ...
- 1,211
3
votes
3
answers
255
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Continuum-distanced complete, ultrametric space
Our professor asked us to find a complete metric space where the intersection of nested closed balls can be empty.
The following space is such an example, and I would like to learn more on it (since ...
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1
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99
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A question about G-Hewitt spaces
In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this ...
- 1,367
3
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1
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249
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Relation between $\mathbb{R}$ and the metric space of bounded functions $f:\mathbb{N}\to\mathbb{N}$
Let $\newcommand{\N}{\mathbb{N}}\newcommand{\B}{\mathbf{B}}\B(\N)$ be the collection of all bounded functions $f:\N\to\N$. (A function $f:\N\to\N$ is bounded if there is $M\in\N$ such that $f(k) < ...
- 48.9k
1
vote
2
answers
202
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Spaces $X$ with every compactification $0$-dimensional with $\beta X\setminus X$ not locally compact
Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications:
Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...
- 1,211
1
vote
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41
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Why does the Kieboom characterization of shape is restricted only to paracompact spaces?
Borsuk founded shape theory as an extension of homotopy theory, appropriate for spaces with bad local properties. Borsuks definition was applied only to compact metric spaces. Later, this was ...
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1
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131
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Strong ultralimits?
I was going through the book Ultrafilters Throughout Mathematics and I came across the notion of ultralimits, defined below.
Ultralimit. Let $(X,\tau)$ be a topological space, $(x_i)_{i\in I}$ be a ...
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2
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314
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Spaces with every compactification $0$-dimensional which aren't locally compact
Recently I've proven the following theorem
Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent:
Every compactification of $X$ is zero-dimensional....
- 1,211
7
votes
2
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383
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Connectivity of fibers under fibration replacement
Assume all the spaces mentioned below are simply connected CW complexes. Let $ f: X \to Y $ be a continuous surjctive map between CW complexes, where $ f $ is not necessarily a fibration. Assume that ...
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9
votes
1
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425
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Delta-generated spaces vs CW complexes
$\newcommand\Top{\mathrm{Top}}\newcommand\CW{\mathrm{CW}}\newcommand\Deltagenerated{\text{$\Delta$-generated}}\newcommand\Spaces{\mathrm{Spaces}}\newcommand\DeltaSpaces{\text{$\Delta$-Spaces}}$I am ...
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3
votes
1
answer
136
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For $\mathbb R^n \times Q \cong \mathbb R^m \times Q $ must $n = m$? ($Q$ is the Hilbert cube)
There are several theorems describing the topology on hyperspaces of convex subsets of $\mathbb R^n$ under the Hausdorff metric. For example Antonyan and Jonard-Pérez prove the space of compact convex ...
- 1,955
1
vote
2
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127
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Homeomorphism and boundary of a complementary component
Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.
More precisely, let $h:X\to X$ be a homeomorphism.
...
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30
votes
2
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2k
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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
8
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1
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198
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Topological property of the space of probability measures
Suppose that $\mathbb{P}$ is the metric space of Borel probability measures on the interval $[0,1]$ equipped with the topology of $w^*$ convergence.
Consider also $\mathbb{P}_{ac}, \mathbb{P}_{s}$ the ...
0
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0
answers
43
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Equivalent conditions for $z$-embeddability
I am looking for where this specific theorem of Blair is originally located:
Theorem. Let $S\subseteq X$, the following are equivalent:
$S$ is $z$-embedded
If $A, B\subseteq S$ are disjoint zero-...
- 1,211
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92
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Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
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2
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297
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Compactly generated and paracompact $\Rightarrow$ Hausdorff?
In A Concise Course in Algebraic Topology by May, a proposition is stated that any open cover of a paracompact space has a numerable refinement, where the space is assumed to be compactly generated ...
9
votes
1
answer
625
views
The reals: a topological lattice in more than the obvious way?
Define a topological lattice as a (not necessarily bounded) lattice in $\textbf{Top}$, i.e. meet and join are continuous maps $X^2 \rightarrow X$. There are two obvious topological lattice structures ...
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2
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1
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103
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LCH spaces $X$ such that if $Y$ is a perfect image of $X$, then $Y$ is zero-dimensional
I am looking for locally compact Hausdorff spaces $X$ with the following property:
If $f:X\to Y$ is a perfect map onto locally compact Hausdorff space $Y$, then $Y$ is zero-dimensional.
One can see ...
- 1,211
7
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1
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339
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Is $C(X, \{0,1\})$ locally compact?
Let $X$ be a locally compact Hausdorff space. Let $C(X, \{0,1\})$ be the space of continuous functions $X \to \{0,1\}$ with the compact-open topology, that is, the topology generated by the following ...
- 1,485
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1
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98
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Is every subgroup closed in this complete, nondiscrete topological group?
Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
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0
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42
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Topologizing quasi orders with regards to products
This morning I was asked by a colleague for the "right" way to construct a topology on a quasi-order (aka preorder, a reflexive and transitive relation) such that the topology on a product ...
- 1,392
4
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1
answer
239
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True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k
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2
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596
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If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
2
votes
2
answers
154
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Closure of $C([0,1]^2)$ via weak*-topology [closed]
Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$.
The dual space of $C([0,1]^2)$, denoted by $C^*([...
- 349
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0
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47
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Are W-spaces with countable pseudocharacter first countable?
Cross-post of a question originally asked by Almanzoris on Mathematics Stack Exchange.
A topological space $X$ is called W-space if P1 has a winning strategy at each point $x \in X$ for the following ...
- 1,392
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1
answer
79
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Dimension of a manifold derived from a dense $G_{\delta}$ subspace
Let $X,Y$ be (compact connected) topological manifolds of dimensions $n,m$, respectively. Assume that a dense $G_{\delta}$ subspace $A$ of $X$ is homeomorphic to a dense $G_{\delta}$ subspace $B$ of $...
7
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2
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488
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Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
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