All Questions
5,185 questions
11
votes
3
answers
890
views
Structure theorems for compact sets of rationals
Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
4
votes
2
answers
164
views
$|\mathsf{RO}(X)|$ vs. $|\tau_X|$ for Tychonoff spaces
Let $\tau_X$ denote the collection of open subsets of a topological space $X$ and let $\mathsf{RO}(X)$ be the subset of $\tau_X$ made up of regular open subsets. With this terminology, the inequality ...
- 1,364
2
votes
1
answer
161
views
Existence of a Borel measurable function
Let $X$ be a compact metric space and $Y\subset X$ be a compact set. Assume that $f_1, f_2: Y \to \mathbb{P}\mathbb{R}^2$ are continuous functions. Let $N \subset \mathbb{P}\mathbb{R}^2$ be a ...
11
votes
2
answers
605
views
Example of an uncountable scattered space with some properties
This might be an easy question, maybe the example I'm looking for is common knowledge. As always, recall that a topological space $X$ is scattered if and only if every non-empty subset $Y$ of $X$ ...
- 42k
3
votes
0
answers
122
views
A space with independent tightness
Recall that the tightness of a topological space $X$ is defined as the least cardinal $\kappa$ such that for every non-closed subset $A$ of $X$ and every point $x \in \overline{A} \setminus A$, there ...
- 4,713
2
votes
0
answers
76
views
Maps defined on the set of Turing degrees
Let $\mathcal{D}$ be the collection of Turing degrees. Are there nontrivial maps $\phi:\mathcal{D}\to \mathcal{D}$ which is natural to consider? For instance, I wonder whether maps which are ...
- 4,459
0
votes
0
answers
42
views
Conditions on a set implying properties on neighborhoods
Suppose $F$ is a closed set in a Euclidean space, and for $\epsilon>0$, let $V_\varepsilon$ be the $\varepsilon-$neighborhood of $F$ i.e. the set of points $x$ having a distance less than $\...
- 411
7
votes
1
answer
354
views
Decomposition of manifolds with toroidal boundary
Let $\mathcal{M}$ be a compact, connected, oriented 3-manifolds with non-empty connected boundary $\partial\mathcal{M}$. Then, following this article, it is stated that $\mathcal{M}$ can be written as
...
- 12.3k
1
vote
1
answer
90
views
Affine semigroup generating a lattice
This is a cross-post from MSE.
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
- 63.9k
7
votes
1
answer
466
views
When is a basis of a topological space a Grothendieck pretopology?
Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more ...
- 37.8k
7
votes
1
answer
350
views
Pushouts of injective monoid homomorphisms
Given a pushout square in the category of monoids
$$\begin{array}{ccc}A & \rightarrow & M \\ \downarrow && \downarrow \\ N & \rightarrow & P\end{array}$$such that $A \to M$ and ...
- 64k
6
votes
0
answers
297
views
Regarding homology of fiber bundle
Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...
- 585
3
votes
1
answer
255
views
Is projection of locally connected compact subset locally connected?
This question was originally at Math Stackexchange, but had no answers:
https://math.stackexchange.com/questions/4348707/is-projection-of-locally-connected-compact-subset-locally-connected
Problem
Let ...
- 42k
3
votes
0
answers
148
views
Topologically characterizing metrizable spaces
There are some well-known theorems that imply that some metrizable spaces, when satisfying other topological properties, are unique up to homeomorphism. Here are a few examples, where "perfect&...
2
votes
0
answers
369
views
Components of the complement of a compact set
Suppose $K$ is a compact subset of $\mathbb{R}^m$ ($m>1$), and $0<r<R$ are fixed numbers. Let $A$ be the set of points having a distance $<R$ and $>r$ from $K$. My questions are
If $K$ ...
CommunityBot
- 1
1
vote
0
answers
46
views
A question related to injective envelope for a system of DGA's
I was trying to read Fine and Triantafillou's paper "On the equivariant formality of Kahler manifolds with finite group action".
They have defined the enlargement at $H$ of a system of DGA's ...
- 1,177
2
votes
1
answer
430
views
General topology book recommendation for advanced probability theory
I would like to know if anyone could suggest a general topology book for a deeper understanding of probability at advanced level. If there is an advanced topology book oriented to probabilists, I ...
- 14.2k
8
votes
1
answer
351
views
Zero-dimensional $F$-space which is not strongly zero-dimensional
Does anyone know of an example of a (Tychonoff) $F$-space which is zero-dimensional but not strongly zero-dimensional?
By an $F$-space we mean every cozeroset is $C^*$-embedded.
By zero-dimensional ...
- 11.4k
5
votes
2
answers
646
views
Can functions be differentiable on sets with empty interiors?
As a simple example, suppose we have a function $f: \mathbb{R}^3 \to \mathbb{R}$ defined on the set (and taking $+\infty$ everywhere else),
$$\{x \in \mathbb{R}^3| x_1 \in [-1, 1], x_2 \in [-1, 1], ...
- 351
12
votes
2
answers
682
views
Non-sequential spaces in the wild
TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
- 16.5k
1
vote
0
answers
67
views
Multisymplectic connections and topological invariants
I was wondering if there exists any notion of multisymplectic connection that naturally generalizes symplectic (Fedosov) connections in symplectic geometry.
From symplectic connections, it is well ...
- 405
2
votes
1
answer
67
views
$E$-separated semigroups
Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.
Observe that $X$ is ...
- 42k
4
votes
1
answer
203
views
Generalized limits in Boolean algebras
Let $\mathbb{B}$ be an infinite $\sigma$-complete Boolean algebra. By $\mathbb{B}^\omega$ we denote the countable product of $\mathbb{B}$ with the coordinate-wise operations. Let us call a ...
2
votes
1
answer
151
views
Is a certain property of a continuous map preserved under "surjectification"?
Let $X$ and $Y$ be compact Hausdorff spaces and let $\varphi:X\to Y$ be continuous with a property that if $A$ is a nowhere dense zero-set in $Y$, then $\varphi^{-1}(A)$ is nowhere dense in $X$. Let $...
- 42k
1
vote
1
answer
190
views
Approximations by compact sub-spaces
Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit
$$\varinjlim_{a\in J} K_a$$
for $J$ a directed set ...
- 12.9k
5
votes
1
answer
806
views
Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?
I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric/Hausdorff spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ (under ...
- 16.5k
6
votes
0
answers
156
views
Topological properties of the dual of differential forms
Notation:
$U \subset R^n$, bounded open set
$D^k(U) = \{ \omega : U \to \Lambda^k R^n : \text{compactly supported and infinitely differentiable \}}$
$D_k(U) = D^k(U)'$ is the topological dual space (...
- 6,367
2
votes
3
answers
562
views
Looking for a reference: $f$-divergences are lower semicontinuous
I know that the weak lower semi-continuity of the KL divergence was proved in [1]. If I remember well, the same property is true for any $f$ divergence (with suitable assumptions on the probability ...
- 345
2
votes
1
answer
455
views
Difference between definitions of continuous action, profinite case
My setting is the following : let $G$ be a topological group and $X$ be a topological space. I have the head filled with two possible definitions for a continuous action of $G$ on $X$.
The first could ...
- 38.6k
3
votes
1
answer
140
views
$i$-weight of a metrizable space
Recall that the $i$-weight of a Tychonoff space $X$, $iw(X)$, denotes the minimal weight of all Tychonoff spaces onto which $X$ can be condensed. A standard fact about this cardinal number is that the ...
- 151
9
votes
0
answers
212
views
Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?
Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones.
We can make $\textbf{Top}$ into a simplicially enriched category as follows:
Given topological spaces $X$ and $Y$,...
- 15.9k
5
votes
2
answers
322
views
Can a punctured $\mathbb{C}P^n$ be a retract of a punctured $\mathbb{C}P^{n+1}$?
In this question we consider the standard inclusion of $\mathbb{C}P^{n}\subset \mathbb{C}P^{n+1}$.
It is well known that $\mathbb{C}P^n$ is not a retract of $\mathbb{C}P^{n+1}$.
What about if we ...
- 6,995
1
vote
0
answers
110
views
Zeroth homology of the complement of a closed set
Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$.
Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? ...
- 411
2
votes
1
answer
167
views
Is $\mathbb R$ with cocountable topology star-$K$-compact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A space $X$ ...
- 2,153
18
votes
7
answers
2k
views
Superfluous definitions
It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.
For if a and b are elements of R, and writing + for the group operation then applying ...
- 4,713
70
votes
28
answers
7k
views
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
- 2,821
21
votes
4
answers
4k
views
Is every locally connected subset of Euclidean space R^n locally path connected ?
This is not actually a question asked by me. But since I do not know the answer, I would love to know if someone here could answer it.
- 4,713
1
vote
1
answer
332
views
Extension of measurable function from dense subset
Let $M$ be a compact riemannian manifold equipped with a geodesic distance and let $\mathcal{B}(M)$ be the borel sigma algebra generated by the geodesic distance. Let $(\Omega,\mathcal{F},\mathbb{P})$...
- 128k
6
votes
2
answers
262
views
Partial orders arising from $l$-spaces
Let $X$ be a $l$-space, i.e. a locally compact totally disconnected hausdorff space, which is not compact. Then $P = \{K : K \subseteq X \text{ compact-open}\}$ is a basis for the topology. Regard $P$ ...
- 2,821
0
votes
1
answer
157
views
Does there exist a star-Lindelöf space which is not star-$L$-Lindelöf?
A space $X$ is said to be star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A ...
- 308
1
vote
1
answer
117
views
Does there exist a starcompact space which is not star-$K$-compact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A space $X$ ...
- 2,153
1
vote
1
answer
152
views
Is the Isbell–Mrówka space $\Psi(\mathcal A)$ with $\lvert\mathcal A\rvert=\omega_1$ starcompact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $\operatorname{St}(\bigcup\mathcal V,\mathcal U)...
- 11.4k
3
votes
1
answer
178
views
Is a certain property of a continuous map preserved under a modification of the topology on the target space?
Let $X$ and $Y$ be Tychonoff (i.e. completely regular Hausdorff) topological spaces and let $\varphi:X\to Y$ be a continuous surjection that also has a property that $\operatorname{int}\overline{\...
- 42k
1
vote
1
answer
172
views
A question about pushforward measures and Peano spaces
Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ ...
- 42k
1
vote
1
answer
250
views
Understanding Kelley's intersection number (Boolean algebras)
It is known that:
Theorem (Kelley, 1959). There exists a finite, strictly positive, finitely additive measure on a Boolean algebra $A$ if and only if $A^+$ is the union of a countable number of ...
CommunityBot
- 1
1
vote
1
answer
389
views
About isotopy of simple close curve
In the Primer mapping class group by farb Margalit. We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if ...
- 28.2k
4
votes
1
answer
259
views
Reference request: large generalized probability measures
I'm interested in references relevant to the following: what is the right generalization, if there is one, of a probability measure that takes on values in an structure of more than continuum size?
I'...
CommunityBot
- 1
0
votes
1
answer
43
views
Exhaustions of product subsets by smaller product subsets
Let $X$ be a compact metric space, $A,B\subset X$ be subsets and $f\colon X\times X\to \mathbb{R}$ a continuous function that is strictly positive on $A\times B$. Do there exist increasing sequences ...
- 486
1
vote
1
answer
323
views
Is the restriction of a projection to a compact subset a quotient map?
Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ ...
- 397
18
votes
4
answers
3k
views
Generalized Stokes' theorem
In the Wikipedia article on Stokes' theorem the following claim is advanced without any references given:
The main challenge in a precise statement of Stokes' theorem is in defining the notion of a ...
- 1,446