All Questions
5,184 questions
1
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91
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A term for a submonoid of a free abelian monoid?
Are there multiple ways of characterising which monoids are submonoids of free abelian monoids?
What free abelian monoids are:
A free abelian monoid $\mathbb N^d$ with $d$ generators (where $d$ is an ...
- 4,943
14
votes
1
answer
581
views
How “disconnected” can a continuum be?
A continuum is a compact connected metrizable topological space.
Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint ...
- 3,011
2
votes
0
answers
68
views
Semigroups related to iterated orthogonal complement
Let $R\subset V\times V$ be a relation on a set $V$. For a subset $S\subset V$, define its orthogonal complement with respect to $R$
as
$$S^l:=\{ x: \forall y\in S\ \ (x,y)\in R\},\ \ S^r:=\{y: \...
- 38.6k
3
votes
0
answers
167
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What is the name of the class of topological spaces with the following property ....?
What is the name of the class of topological spaces with the following property $P$ ?
$X\in P$ iff for any open set $W$ in $X$ and any point $x\in \overline{W}\setminus W$ there is an open set $V$ ...
- 1,101
3
votes
4
answers
681
views
Inductive limit of $\mathbb R^n$s is Hausdorff and second countable?
When dealing with infinite jet bundles, one can consider the topological vector space $\mathbb R^\infty$ obtained by taking the projective limit of the inverse system $(\mathbb R^n,\pi^n_m)$, where $\...
- 42k
1
vote
1
answer
360
views
A question about realcompact spaces
Let $X$ be completely regular space, $\beta X$ be Stone-Čech
compactification of $X$, and $\upsilon X$ be Hewitt realcompactification of $X$.
Then $X\subset \upsilon X\subset \beta X$.
If the ...
- 42k
7
votes
1
answer
358
views
When is a contractible space a retract of the Hilbert cube or $\Bbb R^\omega$?
Which contractible spaces appear as retracts of the Hilbert cube or of $\Bbb R^\omega$ ?
One wants to think that a sufficiently “nice” contractible space is necessarily
a retract of the Hilbert cube ...
- 5,596
12
votes
2
answers
607
views
Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set
It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary set?...
- 13.4k
6
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1
answer
236
views
Two numerical monoids are isomorphic iff they are equal
A numerical monoid (or numerical semigroup) is a submonoid $S$ of the additive monoid $(\mathbb N, +)$ of non-negative integers with the property that the set $\mathbb N \setminus S$ is finite.
It is ...
- 38.6k
3
votes
1
answer
149
views
Fiber dimension formula for compact Hausdorff spaces?
In Algebraic Geometry one has a very useful formula for the dimension of fibers. Specifically I am thinking about a statement of the following form:
Let $C$ be a curve over $\mathbb{C}$, and let $S$...
- 4,786
1
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0
answers
190
views
the Brouwer fixed point theorem for maps rather than spaces
Is there a version for the Brouwer fixed point theorem for maps rather than spaces ?
In other words, for a family of endomorphisms, can the fixed point be chosen continuously, under some assumptions ?
...
- 513
4
votes
1
answer
140
views
Whether a functional which preserves maximum for comonotone functions is monotone?
Let $X$ be a compactum (compact Hausdorff space). By $C(X,[0,1])$ we denote the space of continuous functions endowed with the sup-norm We also consider the natural lattice operations $\vee$ and $\...
- 42k
12
votes
3
answers
852
views
Fixed point theorem for the uncountable power of an interval
Does the Brouwer fixed point theorem holds for the uncountable power $[0,1]^\kappa$ of the interval, $\kappa\geq\aleph_1$ ?
That is, does every continuous endomorphism $[0,1]^\kappa\to [0,1]^\kappa$ ...
- 11.4k
6
votes
1
answer
203
views
Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel?
Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Then we have Theorem 1.40 in Rudin's Real and Complex Analysis, i.e.,
...
- 128k
16
votes
1
answer
481
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Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?
Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \...
- 31.5k
4
votes
0
answers
128
views
An uncountable Baire γ-space without an isolated point exists?
An open cover $U$ of a space $X$ is:
• an $\omega$-cover if $X$ does not belong to $U$ and every finite subset of $X$ is contained in a member of $U$.
• a $\gamma$-cover if it is infinite and each $x\...
- 1,101
32
votes
3
answers
6k
views
Is "compact implies sequentially compact" consistent with ZF?
Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
- 4,713
1
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0
answers
138
views
Category whose morphisms are commutative monoids but not enriched
In a recent investigation, I constructed a category $\mathcal{C}$ with the following property. For objects $X,Y \in \mathcal{C}$, the morphism set $\text{Mor}(X,Y)$ is a commutative monoid with ...
- 161
1
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1
answer
249
views
When are fixed point sets in $T_1$ spaces always closed?
Let $X$ be a topological space, and say that $X$ satisfies the closed fixed point set property if every continuous self-map $f:X\to X$ has fixed point set $\operatorname{Fix}(f)=\{x\in X\mid f(x)=x\}$ ...
- 4,727
11
votes
1
answer
493
views
A topological tree is weakly contractible
Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...
- 7,193
9
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0
answers
160
views
Irreducible subcontinuum without Zorn's lemma
In continuum theory we frequently use the fact that two points in a continuum are contained in an irreducible subcontinuum.
A continuum $X$ is a compact connected metric space. A subcontinuum $K\...
- 3,317
5
votes
1
answer
322
views
Does the Rieger-Nishimura lattice over a subset of $\mathbb{R}^k$ stabilize?
Notation: If $U,V$ are open subsets of a topological space $X$, let us write $U\Rrightarrow V$ for the Heyting operation: the largest open subset $W$ of $X$ such that $U\cap W \subseteq V$ (i.e., the ...
- 32.5k
9
votes
1
answer
1k
views
Are there intuitively clear and not technical proofs of homotopy excision theorem?
The proof given in May's "A concise course in algebraic topology", for instance, is not very involved, but quite technical. Are there less technical, but more "ideologically profound&...
- 12.9k
4
votes
2
answers
453
views
Which topological spaces have a standard Borel $\sigma$-algebra?
Call a topological space $X$ standard Borel if $X$ is standard Borel as a measurable space (equipped with its Borel $\sigma$-algebra), i.e. if there is a Borel isomorphism between $X$ and a Polish ...
- 42k
15
votes
3
answers
3k
views
Making CW-complexes metrizable
$\newcommand\met{\mathrm{met}}$It is a basic topological fact that CW-complexes aren't typically metrizable (they must satisfy a certain local finiteness condition) and the quotient topology is to ...
- 5,596
5
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1
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312
views
"Weird-open" maps in topology
Given topological spaces $X$ and $Y$, we define an open map from $X$ to $Y$ to be a map of sets $f\colon X\to Y$ satisfying the following condition:
For each $U\in\mathcal{P}(X)$, if $U$ is open in $...
- 18.4k
20
votes
1
answer
2k
views
Connected and locally connected, but not path-connected
Allow me to use some non-standard terminology:
A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to X$...
- 4,713
6
votes
3
answers
771
views
Null-homotopy of diagonal map
For a sphere $S^n$, the diagonal map $\Delta:S^n\to S^n\wedge S^n$ sending $x\mapsto x\wedge x$ is null-homotopic. This is the homotopy group $\pi_n(S^n\wedge S^n)=\pi_n(S^{2n})$ is trivial.
I'm ...
- 2,821
0
votes
0
answers
131
views
Cyclic group action and finite invariant set
Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$
Is it true that the ...
- 307
4
votes
0
answers
172
views
Brouwer fixed point theorem for non-Hausdorff spaces
Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?
More particularly, is there a formulation of the Brouwer fixed point theorem
which covers both the standard case of ...
- 2,846
2
votes
0
answers
64
views
A particular generalization of free partially commutative monoids
A trace monoid, or free partially commutative monoid, is one with the presentation $\langle \Sigma \mid a_1b_1 = b_1a_1, \dots, a_nb_n = b_na_n\rangle$. The theory of trace monoids has been well ...
- 21
37
votes
1
answer
1k
views
Does there exist a continuous 2-to-1 function from the sphere to itself?
I am interested in the following question:
Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$?
I suspect the answer is no, but I don't know ...
- 105k
2
votes
1
answer
140
views
Is a Boolean algebra with an order continuous topology a measure algebra?
Assume that $B$ is a complete boolean algebra endowed with a Hausdorff topology, with respect to which all operations on $B$ are continuous, $0$ has a base of full sets (recall that $A\subset B$ is ...
- 3,768
0
votes
1
answer
144
views
Left syndeticity and right syndeticity in nilpotent group
$\DeclareMathOperator\Pf{\mathcal{P}_\mathrm{f}}$Question: Does there exist any reference regarding the study of left and right syndeticity in nilpotent group? More specifically, did anyone introduce/...
- 63.9k
3
votes
0
answers
157
views
Closure of the inverse image under the projection map
Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
- 85
8
votes
1
answer
448
views
Can a Shelah semigroup be commutative?
A semigroup $S$ is called
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\mathbb N$...
- 42k
25
votes
2
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808
views
"All retracts are closed" and "all compacts are closed"
I want to follow the discussion from here concerning about the strength of the separation "all retract subspaces are closed".
(A retract subspace of a topological space $X$ is a subspace $A$ ...
- 42k
10
votes
1
answer
223
views
Is there a connected Hausdorff anticompact space that is countably infinite?
Cross-posted from MSE.
Following Bankston - The total negation of a topological property, a topological space is called anticompact if all its compact subsets are finite. The linked MSE post above ...
- 42k
3
votes
1
answer
155
views
On the Menger property and the Alexandroff duplicate
Recall that a space $X$ is Menger if for each sequence $(\mathcal{U}_n)_{n\in\omega}$ of open covers of $X$, there is a sequence $(\mathcal{V}_n)_{n\in\omega}$ such that, for each $n\in \omega$, $\...
- 1,422
1
vote
1
answer
106
views
Rothberger property and semi-open sets
Here is the definition of a space $X$ to be Rothberger, if for each sequence $(\mathcal{U}_{n})_{n\in\mathbb{N}}$ of open covers of $X$, there exists a sequence $(U_{n})_{n\in\mathbb{N}}$ where $U_{n}\...
- 1,422
1
vote
1
answer
81
views
Is the class of rc-spaces closed under products?
Let $(X,\tau)$ be a topological space. A retraction is a continuous map $r:X\to X$ such that $r$ is the identity on $\text{im}(r)$. We call $S\subseteq X$ a retract of $X$ if there is a retraction $r:...
- 1,422
6
votes
0
answers
255
views
Every Polish space is the image of the Baire space by a continuous and closed map, reference
The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501)
Every Polish space (i.e. every separable ...
- 4,713
6
votes
1
answer
422
views
Transitive homeomorphisms of Erdős spaces
A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense.
Does either of the Erdös spaces $\...
- 3,317
3
votes
0
answers
109
views
"Practical" references on mapping spaces as infinite-dimensional manifolds
I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
- 1,171
1
vote
1
answer
177
views
Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?
Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
1
vote
1
answer
111
views
Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$
Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x_n \in E$ with $n\in \mathbb N$,
$$
x_n \to x \iff [f(x_n) \to f(x) \quad \forall ...
8
votes
3
answers
584
views
"All retracts are closed" as separation axiom
The starting point of this question is the fact that any retract of a $T_2$-space is closed.
Let's say a topological space $(X,\tau)$ is $T_{\textrm{rc}}$ if all retracts of $X$ are closed.
All $T_{\...
- 48.8k
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 ...
- 12.9k
2
votes
1
answer
115
views
A question about a realcompact space and upper semicontinuous function
Nancy Dykes says in the proof of Theorem 3.4 in her article Generalizations of realcompact spaces that by a result of
John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative
upper ...
- 5,596
16
votes
1
answer
502
views
Group actions and "transfinite dynamics"
$\DeclareMathOperator\Sym{Sym}$I have a question about what I shall name here "transfinite dynamics" because it involves iterating a topological dynamical system $G \curvearrowright X$ ...
- 4,265