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Does every proximal outer measure, measure all open sets?

Let $\: \langle X,\delta\rangle \: $ be a separated proximity space. Let $\: \mu^* \: : \: 2^{X} \: \to \: [0,+\infty] \: $ be a proximal outer measure. Let $U$ be an open subset of $X$. Does it ...
user avatar
2 votes
1 answer
407 views

Endomorphisms of degree d on a sphere with infinite fibers on a dense subset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint n-...
Hugo Chapdelaine's user avatar
2 votes
1 answer
422 views

Is every zero-dimensional space with no infinite clopen partition pseudocompact?

For this question, we say that a zero-dimensional space $X$ is $\omega$-pseudocompact if every partition of $X$ into clopen sets is finite. In other words, a zero-dimensional space $X$ is $\omega$-...
Joseph Van Name's user avatar
2 votes
2 answers
439 views

countably complete filters

Is there any description of the set of countably complete filters on the lattice of dense $G_{\delta}$ subsets of a compact, second countable metric space? [I haven't just dreamt this up: it describes ...
Douglas Somerset's user avatar
2 votes
1 answer
214 views

union of Stone-Cech remainders

Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
Douglas Somerset's user avatar
2 votes
1 answer
512 views

Question about analytic curves

Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true ...
Brian Lins's user avatar
2 votes
1 answer
483 views

Whether fine topology and uniform topology on C(X,Y) coincide , when metric on Y is bounded

Whether fine topology and uniform topology on C(X,Y) coincide , when metric on Y is bounded
user17925's user avatar
  • 121
2 votes
1 answer
911 views

A density condition for metric spaces

I have encountered the following property. Can anybody tell me if it already exists in literature and/or is equivalent/similar to other well-known properties? Property: $(X,d)$ metric space. For ...
Valerio Capraro's user avatar
2 votes
1 answer
243 views

Induced pretopologies on sSet

Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a ...
David Roberts's user avatar
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2 votes
1 answer
1k views

Finding saturated open sets

Suppose I have a continuous map $f:X\rightarrow Y$. Then one can wonder, whether for every open set $U\subset X$ the set $U':=\{x\in X|f^{-1}(f(x))\subset U\}$ is open again. This is not true in ...
HenrikRüping's user avatar
2 votes
1 answer
1k views

monoid ring and some structure within it - how is it called?

I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure ...
kakaz's user avatar
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2 votes
0 answers
86 views

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 ...
user39598's user avatar
  • 719
2 votes
0 answers
159 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
  • 649
2 votes
0 answers
92 views

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 ...
Learner's user avatar
  • 141
2 votes
0 answers
29 views

When are canonical maps of a filtered colimit open/closed, given that the transition maps are open/closed?

Let $X_i$ be a filtered diagram of topological spaces. I am interested in when the canonical maps $f_i:X_i\rightarrow \text{colim } X_i$ are open/closed. It is pretty easy to show that if the ...
James's user avatar
  • 41
2 votes
0 answers
146 views

Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R}$

About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the ...
Ali Taghavi's user avatar
2 votes
0 answers
104 views

When do filtered colimits commute with finite products in Top

It is well known that filtered colimits commute with finite products (more generally any finite limit). This is not the case in general in Top due to Top not being cartesian closed. My question is is ...
James's user avatar
  • 41
2 votes
0 answers
71 views

Topological measure theory on spaces that are not completely regular

In the usual discourse regarding approaches to measure theory, it is often pointed out that the restriction of topological measure theory to locally compact Hausdorff spaces is insufficient. However, ...
Cameron Zwarich's user avatar
2 votes
0 answers
81 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
2 votes
0 answers
156 views

Testing for weak homotopy equivalences with compact Hausdorff spaces

Let $f \colon X \to Y$ be a weak homotopy equivalence between topological spaces. If I am not mistaken, then one can rephrase this by stating that the induced map $[K,X] \to [K,Y]$ between homotopy ...
AlexE's user avatar
  • 2,998
2 votes
0 answers
55 views

Fundamental group of cyclic branched cover of affine plane

Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
Doug Liu's user avatar
  • 615
2 votes
0 answers
89 views

Union of two open, open-unicoherent sets whose intersection is connected

I stumbled upon the following proposition, and haven't found an error in my proof yet. By "open-unicoherence" I mean unicoherence with closed sets replaced with open sets in the definition. ...
Calvin Wooyoung Chin's user avatar
2 votes
0 answers
80 views

An alternative definition for finitely generated (and principal) ideals in a semigroup

Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power ...
Salvo Tringali's user avatar
2 votes
0 answers
414 views

$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]

If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$ \left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
Ola Sande's user avatar
  • 705
2 votes
0 answers
35 views

Continuity of Kernel Mean Embeddings

Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
Gaspar's user avatar
  • 161
2 votes
0 answers
91 views

A recursive description of the smallest divisor-closed subsemigroup containing a set

Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid). ...
Salvo Tringali's user avatar
2 votes
0 answers
164 views

Triviality of map $(\Sigma \theta)^*$

We know that there is a cofibration sequence $$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
Sajjad Mohammadi's user avatar
2 votes
0 answers
136 views

Progess on conjectures of Palis

I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures "Global Conjecture: There is a dense set $D$ of dynamics such that any element of ...
NicAG's user avatar
  • 247
2 votes
0 answers
92 views

Explicit CW-complex replacement of the space of reparametrization maps

Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
Philippe Gaucher's user avatar
2 votes
0 answers
104 views

Unordered configuration space with non-distinct points

Consider a topological space $X$, a natural number $n>0$ and the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if there is a permutation $\sigma$ ...
Phil-W's user avatar
  • 1,035
2 votes
0 answers
123 views

Homotopy type of a 3-manifold produced via Dehn surgery?

My apologizes if this is a fairly elementary question, I am still a novice when it comes to 3-manifold topology. I am wondering the following: by Kirby calculus, we know that two links (say in $S^{3}$ ...
Elliot's user avatar
  • 295
2 votes
1 answer
171 views

Is the collapse of a totally disconnected compact Hausdorff space still totally disconnected?

Let $S$ be a totally disconnected compact Hausdorff space and let $A\subset S$ be a closed subset. Let $S/A$ denote the space we get when collapsing $A$ to a point. Is this space still totally ...
user avatar
2 votes
0 answers
176 views

On the origin of power semigroups

Let $S$ be a (multiplicatively written) semigroup. Equipped with the (binary) operation of setwise multiplication $(X, Y) \mapsto \{xy \colon x \in X, \, y \in Y\}$, the family of all non-empty ...
Salvo Tringali's user avatar
2 votes
0 answers
49 views

$\sigma$-compactness of probability measures under a refined topology

Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
Hans's user avatar
  • 195
2 votes
0 answers
57 views

Is a “well-behaved” closed subbasis for the topology generated by a closure operator a closed basis for the closure operator itself?

Let $\Omega$ be a set, $\mathcal{c}: \mathcal{P}(\Omega) \rightarrow \mathcal{P}(\Omega)$ be a closure operator (i.e., $\mathcal{c}$ satisfies $X \subseteq \mathcal{c}(X)$ and $\mathcal{c}(\mathcal{c}(...
David Gao's user avatar
  • 2,830
2 votes
0 answers
92 views

Can this order relation, defined in terms of all topological spaces, be defined in terms of the reals alone?

Let $K$ be the operator monoid under composition of Kuratowski's $14$ set operators generated by topological closure $k$ and complement $c.$ Kuratowski's 1922 paper gives the poset diagram of the ...
mathematrucker's user avatar
2 votes
0 answers
185 views

Properties of universal fibration

I am trying to read the following paper [1] (Becker, James C.; Gottlieb, Daniel Henry Coverings of fibrations. Compositio Math.26(1973)) where the authors mentioned that for any fiber $F$, there ...
gola vat's user avatar
  • 179
2 votes
0 answers
48 views

The world of non-weak*-topologies on $\mathcal{P}(X)$

Let $X$ be a metrizable space and consider $\mathcal{P}(X)$, the set of all probability measures on $X$. Typically, the weak*-topology is considered on $\mathcal{P}(X)$, which is a very natural ...
alhal's user avatar
  • 429
2 votes
0 answers
227 views

Is the product of two outer regular Radon measures outer regular?

Everything is nice on second countable spaces: the product of two outer regular Radon measure is still an outer regular Radon measure. But what happens without the assumption of second countability? ...
Thomas Lehéricy's user avatar
2 votes
0 answers
159 views

Are there hereditarily square-boxed plane continua?

A plane continuum is a bounded, closed and connected subset of the plane. A bounding box $B$ for a plane continuum $C$ is a rectangle $B=[a,b]\times[c,d]$ (including sides and interior) such that $C$ ...
Mirko's user avatar
  • 1,375
2 votes
0 answers
105 views

What is known about sublocales defined by regular nuclei?

(For basic terminology, which is supposed to be standard anyway, see this other question, which inspired this one.) I am interested in nuclei $j\colon L\to L$ on a frame $L$ which are regular elements ...
Gro-Tsen's user avatar
  • 32.5k
2 votes
0 answers
156 views

Do Grothendieck topoi with enough points satisfy the fan theorem internally?

Fourman and Hylland proved in the 80s that all spatial topoi satisfy the full fan theorem internally, while there are examples of localic topoi that do not satisfy it. This leads one to conjecture a ...
saolof's user avatar
  • 1,947
2 votes
0 answers
339 views

Blow up at an ordinary double point

Let $X \subset \mathbb{C}^n$ be a complex complete intersection surface with only ordinary double point singularities. Let $o$ be such an ordinary double point. Let $\tilde{X}$ be the strict transform ...
Serge the Toaster's user avatar
2 votes
0 answers
74 views

Is there a literature name for this concept of a "graded metric"?

Given a space $X$, I have been thinking about a function $d\colon X \times X \times \mathbb{N} \to \mathbb{R}_{\geq 0}$ (i.e. with values that are nonnegative reals) with the properties below. One may ...
user501428's user avatar
2 votes
0 answers
58 views

The graph topologies for powersets

Given a topological space $X$ and a metric space $(Y,d_Y)$, there are a number of topologies one may put on the space $\mathcal{C}(X,Y)$ of continuous functions from $X$ to $Y$. Perhaps the most ...
Emily's user avatar
  • 11.8k
2 votes
0 answers
95 views

References (and a question) on the "fine" topology of powersets

Recently I've been trying to understand powerset topologies better, and came upon the following reference: Frank Wattenberg, Topologies on the set of closed subsets. Pacific J. Math. 68(2): 537-551 (...
Emily's user avatar
  • 11.8k
2 votes
0 answers
67 views

When did derivative algebras first appear?

In the paper "The Algebra of Topology" (Annals of Mathematics, 45, 1944), McKinsey and Tarski proposed derivative algebras (p183) to define the derive set in topology as follows. Suppose $K$ ...
Eugene Zhang's user avatar
2 votes
0 answers
107 views

Existence of a nice-ish topology on the powerset of a topological space

This is a follow-up question to my previous question, Existence of a *really* nice topology on the powerset of a topological space, which, in a few words, asked about whether given a topological space ...
Emily's user avatar
  • 11.8k
2 votes
0 answers
70 views

Niceness properties of quotient spaces by continuous equivalence relations

Given an equivalence relation $R$ on a topological space $X$, there are certain conditions we may ask of $R$ that imply certain well-behavedness conditions on the quotient space $X/\mathord{\sim}_R$. ...
Emily's user avatar
  • 11.8k
2 votes
0 answers
98 views

Closed images of linearly ordered spaces

Is there a description of the class of continuous closed images of linearly ordered spaces?
Smolin Vlad's user avatar

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