All Questions
5,184 questions
11
votes
1
answer
341
views
Density of linear subspaces in $C(K)$
Let $K$ be a compact Hausdorff space and denote by $C(K)$ the space of all real valued and continuous functions on $K$. We endow $C(K)$ with the supremum norm topology, making it a Banach space.
...
- 467
7
votes
0
answers
148
views
Is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?
In short, the question is in the title: is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?
A bit of context:
Given a topological space $X$, a family $\...
- 775
0
votes
0
answers
150
views
Connectedness of deleted symmetric product
Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
- 674
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 ...
5
votes
1
answer
191
views
Monoidal topology and coarse spaces
Is there a description of (quasi-)coarse spaces that is analogous to the description of (quasi-)uniform spaces as lax algebras?
- 3,816
7
votes
1
answer
384
views
Compact Hausdorff spaces as a cocompletion of profinite sets
It is well-known that the category CH of compact Hausdorff spaces has a strong categorical flavor (e.g. Properties of the category of compact Hausdorff spaces, which includes Manes' theorem asserting ...
- 2,040
11
votes
1
answer
428
views
Is the Mandelbrot set Suslinian?
The Mandelbrot set is known to be (path-)connected and compact. A non-degenerate space with these properties is called a continuum.
A continuum $X$ is Suslinian if every collection of non-degenerate ...
- 3,317
1
vote
0
answers
70
views
Is the property $S_1(\Omega,\Gamma)$ preserved when the metrizable separable topology is finer?
Let $(X,\tau)$ be a metrizable separable space and $X$ has the property $S_1(\Omega,\Gamma)$.
Suppose that $(X,\tau')$ be a metrizable separable space such that $\tau\subset \tau'$.
Will space $(X,\...
- 1,101
0
votes
0
answers
97
views
Generator of an analytic semigroup
Perhaps I have a naive question. My question is as follows:
When we consider a Cauchy proposition of the following form:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
- 39
2
votes
0
answers
88
views
Union of two open, open-unicoherent sets whose intersection is connected
I stumbled upon the following proposition, and haven't found an error in my proof yet.
By "open-unicoherence" I mean unicoherence with closed sets replaced with open sets in the definition.
...
5
votes
0
answers
108
views
Structure of well-ordered commutative monoids
Let $(M,+)$ be a commutative monoid. Let $<$ be a well-ordering on $M$, where
$\forall a\in M,\ 0\leq a$
$\forall a,b,c\in M,\ a<b\Rightarrow a+c<b+c$
The first condition means $M$ will be ...
- 18.7k
3
votes
1
answer
132
views
Is it possible to determine whether the critical values are nowhere dense in the case of a bounded set of stationary points?
Let $g:\Bbb R^{d}\rightarrow \Bbb R$ be a non-negative, continuously differentiable function satisfying the following two conditions:
The set $\{\theta\in\Bbb R^n\mid\|\nabla g(\theta)\|<\eta\}$ ...
- 51
1
vote
1
answer
162
views
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
- 3,104
-1
votes
1
answer
115
views
Is this function on the Cantor set continuous? [closed]
Let $S = \displaystyle \prod_{n \ge 1} \{ 0, 1\}$ be the set of binary sequences, so $S$ with the product topology is homeomorphic to the Cantor set. Endow $\mathbb{Z}_{\ge 1} \cup \{ \infty \}$ with ...
- 233
0
votes
0
answers
55
views
Any useful bases for the topology induced by the $t$-Wasserstein distance?
I am working on $\mathbb R ^d$ equipped with the usual Euclidean metric. I know of one nice base for $\mathcal W _t$, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
- 291
0
votes
2
answers
348
views
If a graph embedded on a surface is divided by a curve into a right and left that do not intersect can it be embedded on a surface of smaller genus?
Suppose we have a graph $G$ embedded on a (smooth, orientable etc) surface $Q$. Suppose there is a cycle $C$ of $G$ such that
$C$ does not separate our surface $Q$ into two connected regions and ...
- 111
1
vote
1
answer
132
views
Variants of Dirichlet-type function as a pointwise limit of continuous functions
Problem
Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both ...
- 129
3
votes
1
answer
171
views
Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational
Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
- 10.5k
10
votes
1
answer
395
views
Two dimensional perfect sets
Consider the following family of sets
$$ \begin{align*}
\mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in ...
- 2,286
78
votes
12
answers
12k
views
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
- 2,392
8
votes
1
answer
264
views
Does the continuous image of a disc contain an embedded disc?
Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
- 13.6k
10
votes
1
answer
597
views
Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?
We work in ZFC.
Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$.
A field $E$ is ...
- 1,211
1
vote
1
answer
84
views
Simple convergence of convex compact set implies Hausdorff convergence
I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 125
1
vote
2
answers
131
views
Axiomatic definition of Katětov closure operator
In the book "Categorical Structure of Closure Operators with Applications to Topology" by Dikranjan and Tholen a Katětov closure operator is defined in terms of filter covergence:
$k_X(M):=\{...
1
vote
0
answers
76
views
Uniform approximation over compacts using weighted function spaces
I'm interested in approximations over the so-called weighted function spaces.
Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
- 161
2
votes
1
answer
132
views
The separability of superextensions
The superextension $\lambda X$ of a compact Hausdorff space $X$ is the space of maximal linked systems of closed subsets of $X$, endowed with the Vietoris topology inherited from the double hyperspace ...
- 42k
3
votes
1
answer
268
views
Is the Fortissimo space on discrete $\omega_1$ radial?
Let $\omega_1$ have the discrete topology. Its Fortissimo space is $X=\omega_1\cup\{\infty\}$ where neighborhoods of $\infty$ are co-countable.
A space is radial provided for every subset $A$ and ...
- 1,402
7
votes
3
answers
1k
views
Connected space being not locally connected at each point
Say that a topological space $X$ is locally connected at some point $x$, if it has a local base at that point consisting of connected open sets.
Also $X$ is locally connected if it is locally ...
2
votes
0
answers
80
views
An alternative definition for finitely generated (and principal) ideals in a semigroup
Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power ...
- 10.5k
8
votes
1
answer
351
views
"Compactness length" of Baire space
Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton?
In more ...
- 20.5k
1
vote
0
answers
61
views
Necessary or sufficient conditions for the $k$-fold intersection to be empty in a covering with a "tree structure"
Consider a finite collection of $d$-dimensional balls $\mathfrak{B}=\{B_1,\ldots,B_n\}$ which cover a PL $d$-manifold $M$, i.e. $M=\bigcup_{i=1}^{n}B_i$. Suppose we want to compute the Euler ...
- 159
0
votes
0
answers
101
views
Finding an example if it exists, for a non-contractible and contractible space with special requirement on quotients of their union?
Let $A$ and $B$ be subsets of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, such that $A$ is non-contractible, $B$ is contractible and $B$ is not an one-point set.
I would like to find example(s) ...
6
votes
1
answer
231
views
Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations
Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
- 331
2
votes
2
answers
274
views
Is a simple closed curve always a free boundary arc?
Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points?
For a simple closed curve $\...
- 23
8
votes
1
answer
1k
views
What's the point of a point-free locale?
In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...
- 5,230
3
votes
0
answers
105
views
Finitely generated Banach lattice $C(K)$ and partitions of unity
Let $E$ be a Banach lattice. A Banach sublattice $L$ of $E$ is called finitely generated if there exists a finite subset $F \subseteq E$ such that
$$L = \bigcap \{ \hat{L} \mid F \subseteq \hat L, \, \...
- 467
6
votes
1
answer
173
views
References on semigroup actions
I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994).
I would like to ask for references on semigroup actions on ...
- 339
0
votes
1
answer
78
views
Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K? [closed]
Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components.
Let $U$ be a connected component of $S \setminus K$ and ...
2
votes
0
answers
55
views
Fundamental group of cyclic branched cover of affine plane
Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
- 615
5
votes
0
answers
94
views
When a compact subset of a TVS can be continuously projected on a closed linear subspace?
Let $V$ be a (Hausdorff) topological vector space, $W\subset V$ a closed linear subspace, $X\subset V $ a compact.
(Q):
When there is a continuous map $P:X\to W$ such that $P(x)=x$ for every $x\in X\...
- 60.5k
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 ...
- 1,402
1
vote
0
answers
81
views
"Star" of a CW-complex
Suppose we have a CW-complex $X$ with a 0-cell $e^0$. Is the union of all the cells (of higher dimensions) for which $e^0$ is a boundary point open in $X$?
I don't know if it has a name, but a similar ...
- 11
25
votes
6
answers
2k
views
Are there infinitely many "generalized triangle vertices"?
Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
- 20.5k
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 $C$.
(...
1
vote
1
answer
153
views
For topological torus action, there is a subcircle whose fixed point is the same as the torus
Let $T=\mathbb{S}^{1}\times \mathbb{S}^{1}\times \cdots \times \mathbb{S}^{1}
$ ($n$ times) be an $n$-dimensional torus acting on any topological space $X$.
The group $G$ is said to act on a space $X$ ...
- 1,367
4
votes
0
answers
249
views
Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation
The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
- 2,858
6
votes
0
answers
169
views
Topological spaces for which $w(X)\leq |X|$ holds
Let $w(X) = \inf\{|\mathcal{B}| : \mathcal{B} \text{ is a base for }X\}$ be the weight of topological space $X$.
For metric spaces and locally compact spaces we have inequality $w(X)\leq |X|$.
This ...
- 1,211
5
votes
2
answers
714
views
On the boundary of a simply connected set
Let $U$ be an open simply connected subset of $\mathbb R^2$. Let $x$ be a boundary point of $U$.
Does then there always exist a continuous function $f\colon[0,1]\to\mathbb R^2\setminus U$ such that $x ...
- 128k
2
votes
0
answers
91
views
A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
- 10.5k
8
votes
4
answers
681
views
Uniform density of Lipschitz maps is space of continuous function — for general metric spaces
Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the ...
- 364