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3 votes
0 answers
42 views

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 $\kappa$, ...
David Gao's user avatar
  • 2,830
7 votes
0 answers
184 views

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 ...
Matthew Kvalheim's user avatar
13 votes
1 answer
839 views

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}...
stgo's user avatar
  • 193
5 votes
1 answer
279 views

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. ...
Matthew Kvalheim's user avatar
12 votes
1 answer
384 views

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 ...
Christian Remling's user avatar
7 votes
1 answer
291 views

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{...
Ronnie Pavlov's user avatar
3 votes
1 answer
144 views

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,...
Chris Sanders's user avatar
2 votes
0 answers
157 views

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 ...
rfloc's user avatar
  • 627
3 votes
1 answer
342 views

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 $\...
Dominic van der Zypen's user avatar
9 votes
1 answer
294 views

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 ...
Calvin Wooyoung Chin's user avatar
10 votes
1 answer
657 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 ...
Tyrannosaurus's user avatar
3 votes
1 answer
130 views

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\...
Dominic van der Zypen's user avatar
3 votes
1 answer
90 views

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 ...
Jakobian's user avatar
  • 1,201
1 vote
0 answers
41 views

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 ...
Emo's user avatar
  • 111
14 votes
1 answer
495 views

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 ...
James E Hanson's user avatar
0 votes
0 answers
62 views

Order-convergence and interval topology on ${\cal P}(\omega)/(\text{fin})$

On any poset $(P, \leq)$ we can consider two different topologies that arise directly from the ordering relation. 1) Order convergence topolog $\tau_o(P)$ : By a set filter $\mathcal{F}$ on $P$ we ...
Dominic van der Zypen's user avatar
5 votes
1 answer
92 views

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 ...
Gro-Tsen's user avatar
  • 32.5k
0 votes
1 answer
99 views

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 ...
Mehmet Onat's user avatar
  • 1,367
3 votes
2 answers
340 views

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-...
piper1967's user avatar
  • 1,177
0 votes
0 answers
43 views

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-...
Jakobian's user avatar
  • 1,201
3 votes
3 answers
255 views

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 ...
aleph2's user avatar
  • 637
1 vote
0 answers
68 views

"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 $N_0\partial N_e\Gamma$ ...
Leo's user avatar
  • 11
2 votes
1 answer
131 views

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 ...
Ray's user avatar
  • 23
3 votes
1 answer
247 views

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) < ...
Dominic van der Zypen's user avatar
1 vote
2 answers
127 views

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. ...
D.S. Lipham's user avatar
  • 3,317
3 votes
1 answer
136 views

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 ...
Daron's user avatar
  • 1,955
0 votes
0 answers
42 views

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 ...
Steven Clontz's user avatar
8 votes
1 answer
198 views

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 ...
an_ordinary_mathematician's user avatar
1 vote
2 answers
202 views

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 $\beta X\setminus X$ ...
Jakobian's user avatar
  • 1,201
4 votes
0 answers
47 views

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 ...
Steven Clontz's user avatar
2 votes
1 answer
103 views

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 ...
Jakobian's user avatar
  • 1,201
0 votes
0 answers
32 views

Hausdorff dimension: The dimension of boundary of a set [migrated]

I can't understand the following statement. If (perhaps not closed) set $S$ has dimension $n$, then the boundary could have any dimension from $0$ to $n$. (Could someone give me an example?) If S ...
TianS's user avatar
  • 129
9 votes
1 answer
424 views

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 ...
user39598's user avatar
  • 521
11 votes
2 answers
314 views

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....
Jakobian's user avatar
  • 1,201
0 votes
1 answer
98 views

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 $G$...
Nick Belane's user avatar
7 votes
2 answers
383 views

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 ...
piper1967's user avatar
  • 1,177
1 vote
0 answers
104 views

Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units

For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
M.G.'s user avatar
  • 7,127
0 votes
1 answer
79 views

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 $...
William of Baskerville's user avatar
4 votes
0 answers
97 views

Is there a concept of a map of Grothendieck sites having dense image?

Someone recently asked if one can talk about a map being etale dense just like one can talk about it being Zariski dense. My main question is: has anyone discussed such a notion? On a simple ...
David Corwin's user avatar
  • 15.4k
1 vote
1 answer
91 views

When is a 2-bridge knot hyperbolic?

It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
YC Su's user avatar
  • 605
7 votes
1 answer
339 views

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 ...
Luiz Felipe Garcia's user avatar
3 votes
0 answers
76 views

Can we generalize the Kuratowski Extension Theorem to Souslin spaces?

The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\...
rfloc's user avatar
  • 627
2 votes
2 answers
154 views

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^*([...
tom jerry's user avatar
  • 349
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 ...
Keith's user avatar
  • 591
0 votes
1 answer
230 views

Questions on the compactness of $L_1([0,1]^2)$'s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
tom jerry's user avatar
  • 349
0 votes
1 answer
100 views

Embeddings of pseudo metric spaces into seminormed Spaces

There is a theorem stating that every metric space embeds isometrically into $\ell _{\infty}$. My question: is there a generalized result for pseudo metric spaces embedding isometrically into semi-...
DJ Forklift's user avatar
0 votes
1 answer
100 views

Limit sequence of regular function in $L_1$‘s unit sphere

Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
tom jerry's user avatar
  • 349
0 votes
0 answers
128 views

The smallest dihedral angle of convex polyhedrons

Given a set of points $\{x_{k}\}_{k=0}^{m} \subset \mathbb{R}^n$, is it always possible to find a constant $c=c(m,n)>0$, depending only on the dimension $n$ and the number $m$, such that, after ...
sorrymaker's user avatar
7 votes
2 answers
297 views

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 ...
LuckyJollyMoments's user avatar
-1 votes
1 answer
167 views

Space of distributions on $[0,1]^2$: weakly compact or not?

Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$. Question 1: Does $\mathcal{X}$ ...
tom jerry's user avatar
  • 349

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