All Questions
825 questions
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A ``1-soft'' improvement of the Parovichenko theorem
This is a ``1-soft'' modification of this problem. We start with the necessary definitions.
Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any ...
- 41.9k
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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 $G$...
- 361
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1
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74
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$T_2$-spaces such that the lattices of open sets can be embedded into each other
Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$.
Does this imply that $(X,\tau)\cong (Y,\sigma)$?
user62017
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1
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172
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Can we build a continuous function from "fibers"/preimages defined over a topological base?
I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...
- 273
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643
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A new generalization of the dimension?
During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
- 4,074
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114
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
- 5,529
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605
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Is there a continuous surjection $\omega^\omega\to \mathbb{R}$? [closed]
Let $\omega$ be endowed with the discrete topology, and let $\mathbb{R}$ carry the Euclidean topology. Is there a continuous surjective map $f:\omega^\omega\to \mathbb{R}$?
(I suppose this would ...
- 48.9k
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66
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Question on existence of almost length-minimizing curve in a general domain?
I have the following question: for a general domain $\Omega$ in $\mathbb{R}^n$, is it true that for each pair of points $x,y\in \Omega$, there exists a curve $\gamma$ connecting $x$ and $y$ in $\Omega$...
- 11
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58
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Does the total space of a bundle satisfy the Tietze extension property when the fiber and base space do satisfy this property?
We say that a Topological space $Y$ satisfies the Tietze extension propery, TE property, if in the formulation of Tietze extension theorem "$\mathbb{R}$" can be replaced by $Y$.
Obvioysly the ...
- 356
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208
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A noncommutative analogy of the tube lemma
Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to \mathbb{C}\...
- 356
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1
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225
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Convergence in compact-open topology on the Sierpiński space
Question:
Equip $\{0,1\}$ with the Sierpiński topology $\{\{1\},\{0,1\},\emptyset\}$, let $X$ be a compact metric space, and equip $C(X,\{0,1\})$ with the compact-open topology. Let $\{B_n\}_{n=1}^{\...
- 5,405
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2
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287
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Distinguishable under manifold topology but indistinguishable under the Alexandrov topology
Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.
In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
- 612
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233
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Does every compact countable space contain a non-trivial convergent sequence?
Problem. Does every compact countable space contain a non-trivial convergent sequence?
This question concerns non-Hausdorff compact spaces. An example of such space is any infinite set $X$ endowed ...
- 41.9k
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113
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Finite sets are residual in the Hausdorff space
Let $X$ be a metric space, let $\mathbb{H}(X)$ denote the set of non-empty closed subsets of $X$ with Hausdorff metric which we denote by $d_{\mathbb{H}(X)}$, and let $\mathbb{H}_{\operatorname{fin}}(...
- 5,405
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238
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Minimal totally separated spaces
Let us call a space $(X,\tau)$ totally separated (ts) if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\...
- 48.9k
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2
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545
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Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?
Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?
clearly every closed neighborhood ...
user31967
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1
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403
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When does a power semigroup have a zero, and what can the zero be?
Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$
This operation is ...
- 1,445
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98
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Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?
(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...
- 2,830
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1
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88
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Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$
Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq ...
- 48.9k
-1
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2
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409
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$X$ is Polish and $N$ is countable. Is $N^X$ Polish? [closed]
$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.
$X^N$ is the collection of all mappings from $N$ to $X$. It is ...
- 263
-1
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1
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406
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Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
- 362
-2
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1
answer
389
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Bounded metric spaces with non-surjective self-isometry
A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.
A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
- 48.9k
-4
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1
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328
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Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]
Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold?
NOTE: PLEASE avoid the ...
- 612
-4
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1
answer
412
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A topological groupoid structure on a pair $(X,A)$
Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.
Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding ...
- 356
-8
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1
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351
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Are there overwhelmingly more finite monoids than finite spaces? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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