All Questions
5,184 questions
2
votes
1
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264
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Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?
As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
- 128k
9
votes
1
answer
370
views
G-topological spaces and locales
Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...
- 63.1k
3
votes
0
answers
246
views
"weakly functorial resolution" of quasi-compact T_1 topological space by quasi-compact Hausdorff space
I have an arguably weird question: Let $X$ be a quasi-compact $T_1$ topological space, could there be a construction that takes such an $X$ as input and outputs a surjection
$$X' \to X$$
with the ...
- 619
6
votes
0
answers
155
views
Is there a Lindelof $P$-space which is not discretely generated?
A space $X$ is:
Lindelof if every open cover for $X$ has a countable subcover.
A $P$-space if every $G_\delta$ subset of $X$ is open.
Discretely generated if for every non-closed set $A \subset X$ ...
- 4,413
14
votes
1
answer
571
views
"Scott completion" of dcpo
If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U $ for ...
- 18.4k
5
votes
0
answers
249
views
Aspherical space whose fundamental group is subgroup of the Euclidean isometry group
Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
- 661
0
votes
1
answer
152
views
Name for a monoid on the basis of a vector space?
Is there a name for the structure of a vector space with a monoid defined on its basis?
Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
- 63.1k
0
votes
1
answer
327
views
Can we generalise groupoids to monoid-oids? [closed]
Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories.
Groupoids correspond to small categories where every morphism is an ...
- 48.8k
0
votes
1
answer
109
views
Extending maps from a discrete set to a Stone-Čech compactification while retaining an injectivity condition
For $S$ a set, let $\beta_{\bf2}(S)$ be a compact, totally disconnected space containing $S$ where $S$ in the subspace topology is discrete and $S$ is a dense subspace, and $\beta_{\bf2}(S)$ has the ...
- 11.4k
3
votes
2
answers
285
views
Cut a homotopy in two via a "frontier"
Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$.
(...
- 52.6k
21
votes
1
answer
638
views
Grothendieck group of the Fibonacci monoid
Let's denote the Fibonacci numbers by $F_0=0,F_1=1,F_{n+2}=F_{n+1}+F_n \; \forall n \ge 0$. According to Zeckendorf's theorem, every positive integer can be represented uniquely as the sum of some (at ...
- 28.7k
5
votes
0
answers
131
views
Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?
Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
- 64k
1
vote
1
answer
344
views
Is there anyway to formulate the Alexandrov topology algebraically?
One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set.
Given this, one finds a one-to-one correspondence between ...
- 612
10
votes
0
answers
242
views
Arhangel'skii's problem revisited
One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is ...
- 4,413
8
votes
1
answer
380
views
Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it nontrivially in a disk?
We call an open subset $D\subset X$ of a manifold $X$ an embedded disk, if there exists a homeomorphism $D\cong \mathbb{R}^n$.
The precise formulation of the question in the title is as follows:
Let $...
- 12.3k
1
vote
0
answers
84
views
Is there a standard name for the following class of functions on non-Hausdorff manifolds?
Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
- 5,429
92
votes
3
answers
14k
views
Is every sigma-algebra the Borel algebra of a topology?
This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...
1
vote
1
answer
732
views
Notations for open and closed sets
I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.
- 4,786
9
votes
2
answers
3k
views
Compact Hausdorff spaces without isolated points in ZF
$S$ is uncountable := $\vert\mathbb{N}\vert<\vert S\vert$
$S$ is noncountable := $\vert S\vert \not\leq \vert\mathbb{N}\vert$
$(X,T)$ is a nice space := $(X,T)$ is a compact Hausdorff space ...
- 2,031
2
votes
1
answer
117
views
Primal identity in matrix semigroup
Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product.
We call $s_1\cdots s_k$ an identity index if
$M_{s_1}M_{...
- 12.9k
2
votes
1
answer
141
views
Is there a bound on the number of connected components of a zero set of an integrable function?
If $f$ is a real-analytic function on $[0,1]^n$, and $f$ has finite differential transcendence degree, is there some way to bound the number of connected components of its zero set or the set where it ...
- 1,446
0
votes
0
answers
73
views
Can we construct general counterexample to support the Weak Whitney theorem? [duplicate]
Can we construct an example for the weak Whitney theorem to illustrate the existence of a continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold that cannot be smoothly ...
- 6,367
1
vote
0
answers
131
views
Can we construct a general counterexample to support the weak whitney embedding theorm?
The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$.
...
- 109
3
votes
1
answer
298
views
Pointwise convergence and disjoint sequences in $C(K)$
Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...
CommunityBot
- 1
3
votes
0
answers
200
views
Contractibility of the pseudo-boundary of the Hilbert cube
Let the separable Hilbert cube $Q=\prod_{i=1}^{+\infty}[0,1]$ embed into the real Hilbert space $H=l^2(\mathbb{Z}^+)$, whose coordinate unit vectors are $\{ e_i \}_{i=1}^{+\infty}$, as the subset $\...
- 1,543
0
votes
0
answers
161
views
Gluing faces of n-cube
Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$.
Let $f_0$ and $f_1$ be faces ...
- 4,589
5
votes
2
answers
478
views
Generalization of the concept of a measure
Consider the following generalization of the concept of a measure:
Let $L = (X, \lor, \land, \bot)$ be a semi-bounded lattice.
Let $M = (Y, \bullet, e)$ be a commutative monoid.
An $(L, M)$-measure is ...
- 41.1k
2
votes
1
answer
212
views
A definition of linking number for knots in $S^3$ using chains in $D^4$
I meet this problem when reading Rolfsen's Knots&Links. After giving 8 different definitions of linking number for knots in $S^3$, he left an exercise: Given disjoint PL knots $J$ and $K$ in $S^3=\...
- 225
0
votes
0
answers
70
views
A cellular automaton with an image that is not closed
Let $G$ be a non-locally finite periodic group and let $V$ be an infinite-dimensional vector space over a field $\mathbb{F}$. Does there exist a nontrivial topology on $V^G$ and a linear cellular ...
- 883
22
votes
4
answers
6k
views
Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
An anonymous question from the 20-questions seminar:
Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
- 4,713
2
votes
1
answer
223
views
Is the projective limit $\mathcal{D}(\mathbb{R})$ separable?
Let $\mathscr{D}(\mathbb{R})$ be the set $C_0^\infty(\mathbb{R})$ of smooth functions with compact support endowed with the following topology:
The initial topology with respect to the family maps $(\...
- 6,367
16
votes
2
answers
1k
views
Which topological spaces have the property that their sheaves of continuous functions are determined by their global sections?
I hope I'm using the terminology correctly. What I mean is this: fix $K = \mathbb{R}$ or $\mathbb{C}$ (I'm interested in both cases). Which topological spaces $X$ have the property that for every ...
- 1,446
4
votes
2
answers
312
views
Must US extremally disconnected spaces be sequentially discrete?
Based upon discussion at Math.SE
Consider the property extremally disconnected, for which the closure of any open set remains open.
Frequently, this property is paired with the assumption of Hausdorff....
- 11.4k
3
votes
2
answers
320
views
Topological characterisations of properties of posets
Finite connected partially ordered sets are in bijective correspondence to connected finite topological spaces that satisfy T_0, see for example the Wikipedia article Finite topological space. Here ...
2
votes
2
answers
390
views
Is a compactly generated Hausdorff space functionally Hausdorff?
Question is the title. I suspect the answer is no, without some further conditions (clearly, normal is sufficient). Pointers to counterexamples would be appreciated, but not necessary.
- 1,446
6
votes
0
answers
182
views
Conditions for metrisability
If a normal, first countable space is the union of countably many open metrisable subspaces, must that space be metrisable?
Partial answers, which I proved in the 1980's, include:
(0) The answer is ...
- 5,596
1
vote
1
answer
628
views
Cohomology of the amplitude space of unlabeled quantum networks
I am investigating a particular map from a product of three-spheres to the moduli space of (non-negative, real edge weight) networks. The map in question is
$$f: \smash{\left( \mathbb{S}^3 \right)}^N \...
- 571
3
votes
1
answer
157
views
Embedding of half open half closed $n$-set in $n$-space
Let $n\geq 2$. Set $\Sigma= \{x\in \mathbb{R}^n: 1\leq |x|<2\}$. Assume $h:\Sigma
\rightarrow \mathbb{R}^n$ is continuous and injective.
Question: Must $h$ also be an embedding?
Some thoughts:
$h|...
- 28k
4
votes
2
answers
468
views
Is a local diffeomorphism with nice boundary values a diffeomorphism?
Let $f:\mathbb{D}=\{z\in\mathbb{C}\mid |z|<1\}\rightarrow\mathbb{C}$ be a local diffeomorphism (i.e. an immersion) from an open disk in the plane to the plane.
The only situation I can image ...
- 28k
1
vote
1
answer
80
views
Reference for k-Hausdorff (in terms of compact T2 images)
In Rezk - Compactly generated spaces a k-Hausdorff property is defined, between weakly Hausdorff and unique sequential limits.
On the other hand, a stronger notion of k-Hausdorff between $T_2$ and ...
- 1,392
0
votes
0
answers
303
views
Proof that a first integral is not a constant function
Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions
$$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$
such that all of them are differentiable and ...
- 247
2
votes
1
answer
155
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Variation of concept of a Lusin space
Citing from Wikipedia,
A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space.
Is there a (previously studied) analogous concept of a Hausdorff (...
- 16.4k
14
votes
0
answers
427
views
Which functions have all the common $\forall\exists$-properties of continuous functions?
This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well.
For a ...
- 20.5k
6
votes
1
answer
457
views
Which maps of topological spaces have the right lifting property with respect to all split monomorphisms?
Let $p : X \to Y$ be a continuous map. We say that $p$ has the right lifting property with respect to split monomorphisms if, for every space $B$, and every retract $A \subseteq B$, and for every ...
- 55.9k
0
votes
1
answer
525
views
Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds
$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
- 612
22
votes
2
answers
1k
views
Can a continuous real-valued function on a large product space depend on uncountably many coordinates?
Is there a reasonably well-behaved topological space $X$ (ideally Polish), a set $\kappa$, and a continuous function $g: X^\kappa\to\mathbb{R}$ that depends on uncountable many coordinates?
If $X$ is ...
- 114k
3
votes
0
answers
176
views
The monoid of stably-free modules over integral group rings
Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules.
In studying objects related to Wall’s D2 problem on CW-...
- 187
8
votes
1
answer
181
views
Stone-topological/profinite equivalence for quandles
A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$:
(Q1) ...
- 38.6k
2
votes
1
answer
116
views
On the maximum elements of a numerical semigroup that have order between $n$ and $2n$
Let $S$ be a submonoid of the non-negative integers $\mathbb Z_{\geq 0}.$ If $\mathbb Z_{\geq 0} \setminus S$ is finite, we say that $S$ is a numerical semigroup. Let $S^*$ denote the collection of ...
CommunityBot
- 1
20
votes
2
answers
2k
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Several questions about Gauss's mathematical conception of braids
I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits ...
