All Questions
5,184 questions
10
votes
2
answers
924
views
Has anyone tabulated 2-knots? Would anyone like to try?
I'd love to have a list of 'small' $2$-knots, for some sense of small. It's not clear what one should filter by, but there are two obvious candidates
Write a movie presentation, and count the frames.
...
- 6,367
6
votes
0
answers
246
views
Making the analogy of finiteness and compactness precise
If one asks about the intution behind compact topological spaces, most often one will hear the mantra
“Compactness of a topological space is a generalisation of the finiteness of a set.”
For example,...
- 1,474
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}(...
- 2,830
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 60.5k
1
vote
0
answers
22
views
Weakening compacity hypothesis in multifunctions intersection
Let $X,Y$ be metric spaces, $x^*\in X$
We define two multifunctions $F_1:X\rightrightarrows Y$,$F_2:X\rightrightarrows Y$.
We recall the upper-semi-continuity in Berge's sense :
A multifunction $F:X\...
- 63.9k
5
votes
1
answer
199
views
Is the unit ball of $B(H)$ a Baire space (with the SOT)?
Let $H$ be a Hilbert space, and let $B(H)$ be the set of bounded linear operators $t \colon H \to H$. Recall that we say $t_i \to t$ in the strong operator topology if $t_i \xi \to t \xi$ for every $\...
- 380
0
votes
0
answers
77
views
Completeness of a normed space
We consider the set $\mathcal{PC}([-r,0],X)$
$$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except
for a finite number
of points } t_* \text{ ...
- 39
18
votes
0
answers
1k
views
"Next steps" after TQFT?
(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.)
Recently, I've been ...
- 377
2
votes
0
answers
92
views
Can this order relation, defined in terms of all topological spaces, be defined in terms of the reals alone?
Let $K$ be the operator monoid under composition of Kuratowski's $14$ set operators generated by topological closure $k$ and complement $c.$ Kuratowski's 1922 paper gives the poset diagram of the ...
- 870
4
votes
1
answer
237
views
Mysior plane is not realcompact
Let $X = \mathbb{R}^2$ with $(x, y)\in X$ for $y\neq 0$ isolated and $(x, 0)$ having neighbourhood basis of the form $$U_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup ...
- 1,211
5
votes
1
answer
117
views
Is there an $\varepsilon$-space which is not $k$-Lindelöf?
Crossposted from https://math.stackexchange.com/questions/4717613
An $\omega$-cover $\mathscr U$ of a space $X$ is a collection of open sets so that $X \not\in\mathscr U$ and every finite subset of $...
- 1,101
0
votes
2
answers
199
views
Give an example of a Rothberger space $X$ which has a Lindelöf subspace $Y$ that is not Rothberger
A space $X$ is said to be Rothberger if for each sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(U_n)$ such that for each $n$ $U_n\in\mathcal{U}_n$ and $\{U_n : n\in\mathbb{...
- 290
47
votes
3
answers
3k
views
A metric characterization of the real line
Is the following metric characterization of the real line true (and known)?
A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real ...
- 41.9k
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 ...
- 179
5
votes
3
answers
4k
views
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 ...
- 429
79
votes
5
answers
5k
views
Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?
The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...
- 695
4
votes
0
answers
425
views
Non-triviality of map $S^{24} \longrightarrow S^{21} \longrightarrow Sp(3)$
Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3).
How to show the composition
$$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$
is non-trivial ...
3
votes
1
answer
151
views
Do we have uniformization theorems for fractional dimensional spaces?
The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions.
I’m interested in how it generalizes for fractional ...
- 2,689
3
votes
1
answer
325
views
A detail in Brown's proof of the generalized Schoenflies theorem
Consider a homeomorphic embedding $h:S^{n-1}\times [0,1]\rightarrow S^n$ and denote
$$S^{n-1}_t=h(S^{n-1}\times \{t\}).$$
The generalized Schoenflies theorem states the closure of each connected ...
- 1,225
5
votes
1
answer
198
views
Iterating the dimensional kernel of a metric space
Fix $n\in \mathbb N$. Let $X$ be a separable metric space of (inductive) dimension $n$. Let
\begin{align}
\Lambda(X)&=\{x\in X:X\text{ is $n$-dimensional at }x\}\\ \\
\Lambda^2(X)&=\Lambda(\...
CommunityBot
- 1
0
votes
1
answer
96
views
A question about filterbasis
K. Hardy and R. G. Wood assert that the family in line 4 is a filterbase. I couldn't show it.
- 11.4k
5
votes
2
answers
202
views
Polish space isometric to its hyperspace
For a Polish space $(X,d)$ its hyperspace $(K(X),d_H)$ is also a Polish space. (Here $K(X)$ denotes the set of all nonempty compact subsets of $X$, and the Hausdorff metric $d_H$ is defined by $d_H(K,...
- 4,713
5
votes
3
answers
1k
views
Does the "continuous locus" of a function have any nice properties?
Suppose $f:\mathbf{R}\to\mathbf{R}$ is a function. Let $S=\{x\in \mathbf{R}|f\text{ is continuous at }x\}$. Does $S$ have any nice properties?
Here are some observations about what $S$ could be:
$S$ ...
- 4,217
0
votes
0
answers
63
views
A construction that sort of merges two semigroups to build a new one
Suppose $H$ and $K$ are semigroups and assume without loss of generality that (the underlying sets of) $H$ and $K$ are disjoint. We can then extend the operations of both $H$ and $K$ to a binary ...
- 10.5k
1
vote
0
answers
51
views
Discreteness of $D^{-1}D$ given that $D$ is uniformly discrete
Let $G$ be a topological group with unit element $e$.
We say that $D\subseteq G$ is discrete if for all $x\in D$ there is a unit-neighborhood $U\subseteq G$ such that $x^{-1}D\cap U=\{e\}$. We say ...
- 151
4
votes
1
answer
245
views
Being contained in a compact set
I have a sequential, hereditarily Lindelöf topological space $\mathcal{X}$, and some subset $A \subseteq \mathcal{X}$. I am interested in the following properties:
There is some compact set $B$ with $...
- 41.9k
5
votes
1
answer
167
views
What structure is preserved by pseudo-homeomorphisms of pseudo-Euclidean spaces?
Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb ...
- 4,943
1
vote
0
answers
109
views
Problems Correction of "Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning "' [closed]
Where I can find the problems correction of this book " Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning "
- 11
13
votes
0
answers
818
views
Covering number estimates for Hölder balls
Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
- 5,405
5
votes
1
answer
165
views
Algebraic solutions of polynomial ODEs
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$
\dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n
...
- 91.8k
1
vote
1
answer
98
views
Intersection of (relativized/preimage) measure 0 with every hyperarithmetic perfect set
Given a perfect tree $T$ on $2^{<\omega}$ viewed as a function from $2^{<\omega}$ to $2^{<\omega}$ define the measure of a subset of $[T]$ to be the measure of it's preimage under the usual ...
- 4,201
4
votes
0
answers
350
views
Does a contractible locally connected continuum have an fixed point property?
I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
CommunityBot
- 1
1
vote
1
answer
168
views
Constructing a continuous function with a prescribed preimage
Given a topological space $X$ and a Banach space $V$, I wonder for which open sets $U$ it is possible to construct a continuous function $f: X \to V$ such that $f^{-1}[B(0, 1)] = U$ - or maybe there ...
4
votes
0
answers
231
views
path category and classifying space
Let $\mathbf{Top}$ be the category of topological spaces and continuous maps, and $\mathbf{Cat}$ be the category of small categories and functors.
There is a path functor $\mathcal{P}:\mathbf{Top}\to \...
- 153
1
vote
1
answer
177
views
Identifying a curve on a closed surface of genus 4
The notation is the one used in the attached picture.
Take a closed, orientable surface $\Sigma_4$ of genus $4$, obtained as the identification space of a polygon with $16$ sides in the usual way. The ...
- 25k
5
votes
0
answers
177
views
Do closed subsets of the generalised Cantor space have an analogue of the perfect set property?
For a regular uncountable cardinal $\kappa$, consider $2^\kappa$ with the "less than box topology" (tree topology? Easton/Bounded support topology?) in which basic open sets are of the form $...
- 2,206
3
votes
0
answers
145
views
Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
- 31
0
votes
0
answers
122
views
Is there a name for this condition on a monoid?
Suppose we have a commutative monoid ${\mathcal M}=\langle M,\otimes\rangle$ such that the usual divisibility relation $\leq_\otimes$ given by $a\leq_\otimes b\Leftrightarrow \exists c(a\otimes c=b)$ ...
- 5,429
0
votes
1
answer
115
views
Generalized Triangle Inequality for Snowflakes
Let $p>0$ and consider a metric space $(X,d)$. I have recently come across a problem where the space $(X,d^q)$ provides is natural; where $q>1$. However, the triangle inquality break (i.e. it ...
- 5,405
5
votes
1
answer
288
views
Extreme amenability of topological groups and invariant means
Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it ...
- 4,713
18
votes
1
answer
991
views
Is the Robertson–Seymour theorem equivalent to the compactness of some topological space?
The Robertson–Seymour theorem concerns downwardly closed classes of isomorphism classes of finite undirected graphs. (Am I committing some sin by referring to a class of classes? An isomorphism class ...
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 $C$ ...
- 1,375
3
votes
0
answers
429
views
"Maehara-style" proof of Jordan-Schoenflies theorem?
The highest upvoted answer to this old question Nice proof of the Jordan curve theorem? is a proof by Ryuji Maehara. I personally really liked/appreciated that Maehara's proof is
A) a fairly ...
- 833
4
votes
1
answer
151
views
Three preprints and one manuscript of Tamura on power semigroups
I'm reading Takayuki Tamura's article "On the recent results in the study of power semigroups", pp. 191-200 in Goberstein & Higgins' Semigroups and Their Applications, Kluwer, 1987 (the ...
- 188k
5
votes
1
answer
281
views
Induced maps on hyperspace topologies
If $X$ is a topological space let $2^X$ denote the set of closed subsets. There are multiple topologies one may equip $2^X$ with (in particular, I have in mind the Vietoris, Fell and similar ...
- 63.9k
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?
...
- 324
9
votes
2
answers
551
views
Is every rational sequence topology homeomorphic?
Crossposted from Math.SE 4698387.
In the rational sequence topology, rationals are discrete and irrationals have a local base defined by choosing a Euclidean-converging sequence of rationals and ...
- 1,392
4
votes
2
answers
261
views
Product of locally Borel sets locally Borel
Let $X$ be a locally compact Hausdorff space with a fixed Radon measure (= Borel measure that is finite on compact subsets, inner regular on open subsets and outer regular on Borel sets) $\mu$ . A ...
- 324
5
votes
0
answers
141
views
Under what assumption on a proper map does the preimage of sufficiently small neighborhood is homotopy equivalent to the fiber?
Let $\pi\colon X\rightarrow Y$ be a proper map of topological spaces. Let's assume that both $X$ and $Y$ are paracompact, Hausdorff and locally weakly contractible. Then is it enough to conclude that ...
- 790