All Questions
5,185 questions
2
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1
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236
views
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 ...
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-...
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 $\omega$-...
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 ...
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 [...
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 ...
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
2
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1
answer
911
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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 ...
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 ...
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 ...
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 ...
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 ...
2
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0
answers
159
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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 ...
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 ...
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 ...
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 ...
2
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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 ...
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, ...
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 \...
2
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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 ...
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 $\...
2
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0
answers
89
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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.
...
2
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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 ...
2
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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
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
2
votes
0
answers
35
views
Continuity of Kernel Mean Embeddings
Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
2
votes
0
answers
91
views
A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
2
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0
answers
164
views
Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$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}.$$...
2
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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 ...
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-...
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 $\sigma$ ...
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 $S^{3}$ ...
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 ...
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 ...
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 \...
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}(...
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 ...
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 ...
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 ...
2
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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?
...
2
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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 $C$ ...
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 ...
2
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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 ...
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 ...
2
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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 ...
2
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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 ...
2
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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 (...
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 $K$ ...
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 ...
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 $X/\mathord{\sim}_R$. ...
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?