All Questions
5,185 questions
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173
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Is there a Lusin space $X$ such that ...?
Is there a Lusin space (in the sense Kunen) $X$ such that
$X$ is Tychonoff;
$X$ is a $\gamma$-space ?
Note that if $X$ is metrizable and a $\gamma$-space then it is not Lusin.
In mathematics, a ...
- 1,101
2
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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: \...
- 513
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
2
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answers
174
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Concrete description of “DeMorganian” open sets
Let me begin with a few definitions. My question will be basically how to simplify them to something more manageable. The motivation for these definitions is given at the end.
Let $X$ be a ...
- 32.5k
2
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0
answers
73
views
What should I call a log scheme with free reduced monoids?
This is a terminology question about a class of log varieties.
Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a ...
- 7,962
2
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answers
220
views
Presentation complex and arbitrary $2$-dimensional CW-complex with same fundamental group
Given a finite group $G$, consider a presentation $P$ of $G$ and consider $X_P$, the presentation complex. Now let $Y$ be any $2$-dimensional CW-complex with $\pi_1(Y)=G$. Is there any relation ...
- 179
2
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161
views
Embedding a monoid into a group via its monoid ring
Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...
user491484
2
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0
answers
52
views
Can we decompose an increasing net of functions into two increasing nets with prescribed supports?
Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...
- 5,529
2
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answers
66
views
Separating property of a finite union of topological disks
Let $X$ be a topological $2$-sphere. Let $D_1, D_2, \dots, D_n \subset X$ be a finite family of closed topological disks (i.e. sets homeomorphic to the closed unit disk). Let $\mathcal{U} = \bigcup_{1 ...
- 111
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103
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$n$-connected spaces (terminology)
A graph is called $n$-connected if it remains connected after removal $\le n$ vertices.
Question. What is the name of an analogous property of topological spaces: a space that remains connected after ...
- 42k
2
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0
answers
181
views
So many types of subwords! How are they called?
Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
- 10.5k
2
votes
0
answers
134
views
Subsets of $\mathbb{S}^n$ fixed by an orientation-reversing self-homeomorphism — Part 1
Call a subset $Z$ of $\mathbb{S}^n$ ambiently-reversible, if there is an orientation-reversing self-homeomorphism $h: \mathbb{S}^n \to \mathbb{S}^n$ fixing $Z$ pointwise.
Question 1: Which subsets of ...
- 1,936
2
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1
answer
443
views
Possible characterisation of compactly generated weakly Hausdorff spaces
Is it true that, in the category $\mathbf{Top}$ of topological spaces and continuous maps, the compactly generated weakly Hausdorff spaces are precisely the spaces arising as filtered colimits of ...
- 1,638
2
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0
answers
73
views
Why does normality imply that a countable base $B$ contains at least one set $U$ whose closure is a subset of another set $V \in B$?
I'm reading Aliprantis and Border's excellent text, Infinite Dimensional Analysis: A Hitchhiker's Guide (PDF available at link, assuming I've done this properly), and I've reached an impasse in the ...
- 21
2
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0
answers
75
views
Dual space induced by a finer topology
Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two seminorms on a space $E$ such that $\|\cdot\|_2\geq\|\cdot\|_1$. Let further $E_i:=(E,\|\cdot\|_i)$ and
$$C_b(E_i):=\{f : E\rightarrow\mathbb{R}\mid f \ \...
- 463
2
votes
0
answers
61
views
Space of continuous paths up to strict reparametrization
Take a Hausdorff topological space $X$. Take two distinct points $x$ and $y$ of $X$. Consider a set $U$ of continuous paths $p$ from $[0,1]$ to $X$ equipped with the compact-open topology such that: $...
- 3,901
2
votes
0
answers
101
views
Concrete topological objects and notions in the category of locales
I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. ...
- 121
2
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answers
130
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Attaching a 2-handle to a once-twisted unlink in the boundary of the 4-ball
Consider the 3-sphere $S_3$ with an unlink loop $L$ whose tubular neighborhood is identified with the solid torus $B_2\times S_1$ with one twist, i.e., such that the image of $x\times S_1$ (where $x$ ...
- 3,001
2
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0
answers
82
views
Is every first-countable symmetrizable space perfect?
Let us recall that a symmetric on a set $X$ is any function $d:X\times X\to[0,\infty)$ such that
for every $x,y\in X$ the following two conditions are satisfied:
$d(x,y)=0$ if and only if $x=y$;
$d(...
- 42k
2
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0
answers
67
views
Type of numerical semigroups is not bounded when embedding dimension is $\geq 4$
I am currently studying numerical semigroups. I know that there is no upper bound for the type of a numerical semigroup with embedding dimension greater or equal than $4$. There is a famous example by ...
- 121
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0
answers
149
views
Polynomial entropy of topological dynamical systems
For a continuous mapping $\varphi: X \to X$ on a compact Hausdorff space $X$ we define the entropy as follows:
Given open covering $\mathcal{U}$, $\mathcal{V}$ of $X$, we call $\mathcal{V}$ a ...
- 467
2
votes
0
answers
76
views
Equilibrium for a game with mixed strategies on a compact ultrametric space
Let $(X,d)$ be a compact ultrametric space. Hartig and de Vink considered the following ultrametric on the set $P(X)$ of probability on $X$:
$$\hat d(\mu,\nu)=\inf\{r>0:\forall x\in X\;\;\mu(B_r(x))...
- 4,535
2
votes
0
answers
73
views
Separately continuous functions of the first Baire class
Problem. Let $X,Y$ be (completely regular) topological spaces such that every separately continuous functions $f:X^2\to\mathbb R$ and $g:Y^2\to \mathbb R$ are of the first Baire class. Is every ...
- 4,535
2
votes
0
answers
108
views
Left-elements of a numerical semigroup generated by two elements
A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
- 31
2
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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
2
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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$ ...
- 411
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0
answers
201
views
A question about infinite product of Baire and meager spaces
Proposition 1: For any space $X$ and an infinite cardinal $\kappa$, the product $X^{\kappa}$ is either meager or a Baire space.
Does anyone have any suggestions to demonstrate Proposition 1?
I was ...
- 561
2
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0
answers
58
views
Dimension changes from global to local immersion
From Hatcher Corollary A.10. the (global) immersion for an $n-$dimensional CW complex is possible in some $\mathbb{R}^N$. I have started with $M(G,n)$ (Moore space of type $(G,n)$, $G$ is cyclic ...
- 1,177
2
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0
answers
198
views
A generalisation of closed and bounded subsets of non-Archimedean fields to topological spaces
The definition of compactness in topological spaces generalises the notion of a subset of $\mathbb{R}^n$ being closed and bounded, as expressed by the Heine-Borel Theorem.
In finite-dimensional vector ...
2
votes
0
answers
129
views
Single theorem for hybrid of winding number and rotation number?
I am trying to make mathematical sense of some observations from my physics research, so I hope that you will bear with me.
For a complex-valued function $z(t)$ dependent on parameter $t$, I calculate ...
- 121
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0
answers
74
views
Is the reversibility of inflation of a subset equivalent to its smoothness?
$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$.
Definition. Let $M \subset \mathbb{R}^n$.
$D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
$Int_r (M): = \{x ~|~ D_r(x) \subset M\}$
...
- 6,962
2
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0
answers
61
views
Does absolute retract imply convex structure?
In the theory of selection, it is known that any compact absolute retract (AR) carries a convexity structure defined by E. Michael. It is also known that a convex structure
developed by Van de Vel ...
- 2,083
2
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0
answers
58
views
A generalization of metrics taking values in partial orders
I'm investigating the origin of the following notion:
Let $S=(S, +, <, 0)$ be a partially ordered semigroup with minimum $0$ (such that $<$ is invariant by the action of $+$ on both sides).
A $S$...
- 775
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0
answers
201
views
are acyclic fibrations of nice spaces absolute extensors for perfectly normal spaces?
A space $Y$ is called an absolute extensor for normal spaces (also sometimes solid) if, for any normal space $X$, closed subset $A$ of $X$, and map $f:A\to Y$, there exists a map $f′:X\to Y$ such that ...
- 317
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0
answers
73
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Nonzero idempotents in compact semitopological semigroups with zero
Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem.
Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
- 2,605
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answers
145
views
Semigroup ideals of a ring or an algebra
Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
- 2,605
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answers
84
views
On the (Brouwer-Koebe) Continuity Method
The so-called Continuity Method is a simple yet powerful method to show that a given continuous injective map is surjective. Namely, suppose that $f:X \to Y$ is a map between two connected manifolds $...
- 2,154
2
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0
answers
119
views
The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
- 4,943
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answers
126
views
(Local) simple connectedness of irreducible algebraic varieties
Let $\mathbb k$ be an algebraically closed field of characteristic zero.
I have two questions:
(1) Is an irreducible algebraic variety $X/\mathbb k$ of dimension at least 2 locally simply connected?
(...
- 129
2
votes
0
answers
222
views
Complete reducibility, in linear algebra and in topology
I thought that this is a simple question and asked it at the Mathematics StackExchange, but I now have to elevate it to MathOverflow.
Consider a representation $A(G)$ of a group $G$ in a vector space $...
- 603
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0
answers
203
views
Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
- 24.9k
2
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86
views
Existence of a proper Morse function
I started with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded (local embedding) it inside $R^{2n}$. Now take a regular neighbourhood $U$ of $X$ in $R^{2n}$ which has the same homotopy ...
- 1,177
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0
answers
67
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A polar open set in a topological subspace?
Suppose $U$ is a bounded open set in $\mathbb{R}^m$ with ($m\geq2$). Is it possible to have a non-empty set $E$ in the boundary $ \partial U$ of $U$ that is open in $ \partial U$ and is polar?
A set $...
- 411
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0
answers
165
views
Dimension of Cartesian products
Is there a notion of dimension such that for all Borel sets $A,B\subseteq\mathbb{R}^{n}$ we have
$$ \dim(A\times B)=\dim(A)+\dim(B)?$$ For topological, Minkowski, packing and Hausdorff dimension this ...
- 1,648
2
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0
answers
162
views
Banach–Mazur game and mappings
The Banach-Mazur game on a nonempty space $X$ is defined as follows: two players, $I$ and $II$, alternately choose nonempty open sets
\begin{matrix}
I & U_0 && U_1 && \cdots ...
- 115
2
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0
answers
74
views
Terminology and notation for generated subgroups
I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
- 12.5k
2
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0
answers
129
views
When every connected component is path connected
Let $X$ be a compact $T_0$ topological space whose connected components are path-connected. Is there any characterization for such a space?
- 71
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0
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83
views
Increasing a nowhere dense set in $\mathfrak E_{\mathrm{c}}$
Let $X$ be a closed nowhere dense subset of the complete Erdos space $$\mathfrak E_{\mathrm{c}}=\{x\in \ell^2:x_n\notin \mathbb Q\text{ for all }n<\omega\}.$$
Can you always find a closed nowhere ...
- 3,317
2
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0
answers
141
views
Example of compact fiber bundle with noncompact fibers
This is a cross post of MSE post somehow:
Is there any example of compact fiber bundle $E$ with noncompact fibers $F$?
Obviously if the base space $B$ is $T_1$ then there is no such example.
- 4,195
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0
answers
55
views
Where can I find literature regarding cardinal invariants of a function space $C(X, Y)$ endowed with the Uniform or Fine topology?
I am working on Function Spaces as a topological space. I want to get a sample paper which studies the cardinal invariants on the function space $C(X, Y)$ rather than on $C(X)$.
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