All Questions
5,185 questions
2
votes
1
answer
131
views
Shrinkable decompositions with uncountably many non-degenerate elements?
Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/_{\mathcal D}$ ...
- 312
1
vote
0
answers
40
views
A exemple of a strongly-continuous contraction semigroup : how to prove the contraction?
I am trying to prove that $P_t := e^{\lambda t (P-I)}$ (where $Pf:= \int f(y) P(\cdot , dy)\in \mathcal{C}_0(\mathbb{R}^d)$, for $f\in \mathcal{C}_0(\mathbb{R}^d)$, $P$ being a probability kernel), is ...
- 111
7
votes
0
answers
266
views
Remote points in $\beta X$
It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space $...
- 79
4
votes
2
answers
233
views
Which Hyperspace Topologies Yield Topological Lattices?
At least on a continuum, the binary operations of intersection and union are Vietoris-continuous. But the Vietoris topology only applies the the collection of NONEMPTY closed subsets, and this means ...
- 1,955
2
votes
1
answer
294
views
Finitely generated ordered monoids and noetherian subsets
(This question was asked a long time ago on MSE but got no answer so far...)
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...
- 6,700
3
votes
0
answers
198
views
Properties of convergence at points of continuity
Let $J$ denote the set of functions $f : [0, \infty) \to \mathbb{R}$ that are right-continuous and have left-hand limits (r.c.l.l.) and such that their points of discontinuity are jumps.
Then $J$ is a ...
- 1,773
1
vote
1
answer
160
views
Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimensional Euclidean spaces
For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple ...
- 6,215
2
votes
0
answers
439
views
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
1
vote
0
answers
134
views
Inverse limits of the interval with a single bonding map below the identity
My question is as follows.
QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f\...
- 6,548
5
votes
1
answer
152
views
Infinite Hausdorff space that is not homeomorphic to any proper quotient
Let $S$ be a set and $\vartheta$ be an equivalence relation on $S$. We say that $\vartheta$ is proper if there are $x\neq y\in S$ with $(x,y)\in\vartheta$.
Is there an infinite Hausdorff space $(X,\...
- 48.9k
8
votes
1
answer
1k
views
Topological necessary and sufficient condition for tightness
Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:
For each $\varepsilon>0$, we can find a compact subset $K$ of $X$...
- 3,993
7
votes
2
answers
766
views
Question about 0-dimensional Polish spaces
Hello everybody,
I'm stuck with proving (or disproving) the following statement.
Statement:
For every $0$-dimensional Polish space $(X,\mathcal{T}\ )$, and a countable basis of clopen sets $\mathcal{...
user11618
-9
votes
1
answer
2k
views
Filters and intersection of two binary relations
Let $\mathfrak{F}$ is the complete lattice of filters (including the improper filter) on some set, ordered
inverse to set-theoretic inclusion.
I will denote $\left\langle f \right\rangle \mathcal{X} =...
- 765
6
votes
1
answer
155
views
Countable subcover of half-open cylinders
While preparing a lecture on dynamic programming principle in optimal stochastic control after the book of Touzi, I discovered a gap in the proof of DPP (page 28 of the book).
Here I simplify the ...
- 1,533
4
votes
3
answers
480
views
closed meagre sets
A closed meagre subset of $[0,1]$ is either countable or homeomorphic to the Cantor set: either way it is $0$-dimensional.
Q.1. Is every closed meagre subset of an $n$-dimensional locally compact ...
- 607
9
votes
1
answer
1k
views
Topology, the board game
Edit: I am reposting this question fom math.stackexchange.com; there may be some professors here who have more experience teaching topology.
This is a math education question that I've been thinking ...
2
votes
2
answers
308
views
Convexity Theorem of Hamiltonian actions - the connectedness part
Suppose we have a Hamiltonian action of a torus $T = T^m = R^m/Z^m$ on a compact, connected symplectic manifold $M$. According to the convexity theorem, we know every fiber of the momentum map $\mu: M\...
- 377
1
vote
0
answers
81
views
A consecutive resolution of continum algebras to a simple continum algebra
Motivated by classical Gelfand Naimark duality, the correspondence between the category of commutative $C^{*}$ algebras and the category of locally compact Hausdorff spaces, we ...
- 366
5
votes
2
answers
2k
views
Can the graph of a continuous function be a rotation of the graph of a discontinuous function?
Can there exist two functions $f,g: \mathbb R \to \mathbb R$ so that $f$ is continuous, $g$ is discontinuous, and their graphs $\Gamma_f, \Gamma_g \subseteq \mathbb R^2$ are related by an isometry? (I ...
4
votes
1
answer
456
views
Homotopy groups of K3
Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface.
Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ ...
2
votes
0
answers
61
views
Looking for a weakly Lindel\"of Tychonoff Moore non-ccc space
Is there a weakly Lindel\"of Tychonoff Moore non-ccc space?
Note that here ccc denotes the countable chain condition; a space $X$ is called weakly Linde\"of if for any open cover $\mathcal U$ of $X$ ...
- 621
5
votes
2
answers
931
views
$2^{\omega_1}$ separable?
I was rereading an answer to an old question of mine and it included a reference to the fact that $2^{\omega_1}$ was separable. I'm having a hard time finding a reference for this fact, and the proof ...
- 1,331
6
votes
1
answer
475
views
Constructible subset of constructible set
Let $X$ be a topological space. Let $F \subset E \subset X$ be subsets. Assume that $E$ is constructible in $X$ and that $F$ is constructible in $E$. Is it true that $F$ is constructible in $X$?
We ...
user45795
12
votes
0
answers
219
views
Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?
Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table.
Let
$$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$
for all $n\in\mathbb{N}$.
Then since $C_{n}$ is a ...
3
votes
4
answers
514
views
Better terminology than "equivalence class of functions"
Let $X = C(\mathbb R)$ be the Fréchet space of real-valued continuous functions. For each $f \in X$ and each compact set $D \subseteq \mathbb R$, let $$[f]_D = \{ g \in X : \mbox{$g(t) = f(t)$ for ...
- 8,512
7
votes
2
answers
394
views
When does a homeomorphism split into essentially minimal homeomorphisms?
Background
Suppose $X$ is a compact metric space, and that $\varphi: X\to X$ is a homeomorphism of $X$.
We say a subset $A$ of $X$ is $\varphi$-invariant if $\varphi(A) = A$. A $\varphi$-invariant ...
- 1,023
2
votes
0
answers
82
views
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
4
votes
1
answer
2k
views
Lebesgue measure of boundary of Caccioppoli set
Can anything be said about the measure of the topological boundary of a Cacciopoli set in $R^n$? Of course, the reduced boundary has finite (n-1)-dimensional Hausdorff measure, but this does not say ...
- 320
2
votes
1
answer
1k
views
Extending uniformly continuous functions on subspaces to non-metrizable compactifications
I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$.
Furthermore, I do have a uniformly continuous function $f$ on $X$. So ...
- 2,998
4
votes
1
answer
310
views
Free action of $\mathbb{Z}(2^{\infty})$ on a compact space
Assume that $X$ is a Hausdorff compact space such that $\forall n\in \mathbb{N}$, we have a free action of $\mathbb{Z/{2^{n}}\mathbb{Z}}$ on $X$. Must $\mathbb{Z}(2^{\infty})$ act freely on $X$?...
- 366
3
votes
0
answers
867
views
The inductive and projective limits of compact Hausdorff topological groups
Are there conditions known under which the inductive or projective limit of a family of compact Hausdorff topological groups is compact? (For instance, such a result is valid for the projective limit ...
- 5,407
24
votes
0
answers
918
views
The topologies for which a presheaf is a sheaf?
Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.
Suppose that $Q$ is a presheaf on $...
- 8,669
6
votes
4
answers
765
views
On Pseudo-finite topological spaces
We recall that a topological space $(X,\tau)$ is Pseudo-finite, if each compact subset of $X$ is finite.
One of the classical example of Pseudo-finite topological spaces can be considered as an ...
- 1,788
5
votes
1
answer
216
views
Continuity of taking collapse maps
Let $U$ and $V$ be open subsets of $\mathbb R^n$ and let $\mathrm{OEmb}(U,V)$ denote the space of open embeddings of $U$ into $V$ with the compact-opent topology. Let $\bar{U},\bar{V}$ denote their ...
- 666
0
votes
2
answers
148
views
Continuous image relation on topological spaces
Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by
for $X, Y \in \text{...
- 48.9k
2
votes
0
answers
62
views
Extensions of an ideal-theoretic criterion for a monoid to be BF
Let $H$ be a multiplicatively written, commutative monoid. We denote by $H^\times$ the set of units (or invertible elements) of $H$, and by $\mathcal A(H)$ the set of atoms (or irreducible elements) ...
- 10.5k
8
votes
2
answers
875
views
Is the mapping cylinder of a Serre fibration also a Serre fibration?
If we have a Serre fibration $p: E \rightarrow B$ with fiber of homotopy type $S^{k-1}$, then we can create a fibration with contractible fiber by first taking the mapping cylinder $M_p$ of $p$ to get ...
- 1,207
4
votes
1
answer
1k
views
Is "second-countable implies separable" equivalent to the Axiom of countable Choice?
It is well-known that a secound-countable topological space is separable. The proof goes like this: Let $(B_n)$ be a (at most) countable base for the topology. We may assume that $B_n$ is nonempty for ...
- 63.1k
4
votes
3
answers
674
views
Is there a (standard) name for $\bar{A}\setminus A$?
This is a notation question:
If $A$ is a set in a topological space and $\bar{A}$ is its closure, is there a (standard) name for $\bar{A}\setminus A$?
- 2,882
0
votes
1
answer
384
views
Heisenberg group acts on the circle
Let $H$ be a Heisenberg group, i.e.
$$
H=\left\langle a,b,c |[a,b]=c,[a,c]=[b,c]=1\right\rangle.
$$
$H$ acts on the circle by homeomorphism which preserves the orientation. If the rotation number of $...
- 21
1
vote
0
answers
293
views
Examples of value quantales
In his paper "Quantales and continuity spaces" R. C. Flagg gives the following examples of value quantales: the lattice $\bf{2}$ of truth values with usual addition, the lattice $\mathbb{R}_{+}$ of ...
1
vote
0
answers
80
views
Extending maps to disc homeomorphisms isotopic to the identity
Consider the closed unit disc $\mathbb D^n$ in $\mathbb R^n$ and its closed subdisc $D$ centered at the origin with radius $1/2$. Denote by $V$ the interior of $\mathbb D^n$. I wonder whether the ...
3
votes
0
answers
104
views
A link of four 2-tori $T^2$ in $S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary with their three $S^1$ boundaries of $T^3$ cyclic permuted to obtain a new 4-...
- 10.5k
3
votes
0
answers
106
views
A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold:
$$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
- 10.5k
1
vote
1
answer
566
views
Example of a topological space
In my recent research, I defined a topological space $X$ to be an $EZ$-space if for every open subset $A$ of $X$, there exists a collection $\{A_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such ...
- 192
11
votes
1
answer
949
views
Magma "actions" (or alternatively, "What is the Yoneda lemma for magmas?")
Arguably the most import thing about groups, semigroups and more generally categories, is that they can act on sets (or even collections of sets in the case of a category). This is the basis for all ...
- 2,392
1
vote
1
answer
95
views
Neighborhoods with proper multiplication
The following question was originally asked here, by C. Dubussy: https://math.stackexchange.com/questions/1802111/neighbourhoods-with-proper-multiplication
Assume we have two closed subsets $F$ and $...
user75246
-1
votes
1
answer
278
views
Decomposition space of $\mathbb{C}$ by concentric circles [closed]
What are the topological properties of the quotient space $X$ obtained from $\mathbb{C}$ by identifying points of the same modulus? I.e., the space $X=\mathbb{C}/E$ where $E$ is the equivalence ...
- 1,704
0
votes
0
answers
120
views
A topology on the product space of Euclidean space and smooth functions space
I'd like to know if there is a well-known topology on the space $S := \mathbb R \times C^\infty(\mathbb R)$, such that $(x_n, f_n) \to (x, f)$ in $S$ with respect the topology is equivalent to
$$(x_n,...
- 1,399
3
votes
1
answer
233
views
Mean on compact metric spaces
Let $X$ be a compact metric space. A $k$ mean on $X$ is a continuous map $f:X^{k}\to X$ which is identity on the diagonal and is invariant under all $k$-permutations. For details, See the following ...
- 366