All Questions
5,185 questions
2
votes
1
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70
views
Equicontinuity-like property of a convex compact set
Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...
- 5,529
2
votes
1
answer
104
views
When almost all points are not isolated in all subspaces
Let $X$ be a compact (non Hausdorff) $T_0$ topological space such that for any subset $\mathcal{A}=\{\mathfrak{x}_\alpha\}_{\alpha\in \Lambda}$
of distinct element of $X$ the set $\{\mathfrak{x}_\beta\...
- 21
2
votes
1
answer
177
views
The Borel class of a subset of $\mathbb Z^\omega$
Define $F(t)=\ln(t+1)$ for $t\geq 0$.
For each sequence of integers $ s=s_0s_1s_2...\in \mathbb Z^\omega$ define $$t^*_{ s}=\sup_{n\geq 0}F^{n}(|s_n|)$$ where $F^{n}$ is the $n$-fold composition of $F$...
- 3,317
2
votes
1
answer
189
views
simplicial complex of two covers
Given two covers $\{U_a,U_b,\dots\}$ and $\{V_1,V_2,\dots\}$ of a space $X$, what is the appropriate idea of simplicial complex? As far as I see there are two ideas, and I was wondering where these ...
- 1,143
2
votes
1
answer
302
views
Density of continuous functions to interior in set of all continuous functions
Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...
- 5,405
2
votes
1
answer
190
views
About product of Baire spaces and forcing
Let $\mathbb{P}=\langle P, \leq \rangle$ be a p.o.
Two elements $p$ and $q$ of it are called
compatible if there is an $r \in \mathbb{P}$ such that $r\leq p$ and $r \leq q$; otherwise they
are called ...
- 561
2
votes
1
answer
172
views
Question about almost locally ccc and the Krom space
Definition 1. A family $\mathcal{B}$ of non-empty open sets in a topological space will be called $\pi$-base (or pseudo-base) if every non-empty open set contains at least one member of $\mathcal{B}$. ...
- 561
2
votes
1
answer
142
views
Explicit construction of a convex metric
Let $(X,d)$ be a compact, connected, locally connected, locally compact metric space.
A result of Bing and Moise (independently) states that $(X,d)$ admits a topology preserving convex metric i.e., ...
- 998
2
votes
1
answer
494
views
Topology: what defines (non-trivial) paths as being the same trace (curve)?
I posted this originally at MathSE but haven't had any feedback and hope for better luck here.
I think I know the answer but can't prove it.
Assume that $Y$ is a Hausdorff space and firstly that $p,...
- 165
2
votes
1
answer
153
views
Define a homomorphism of a set of graphs to its power set
Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G_1$ and $G_2$ is,
$G_1\cup G_2$
$=\langle V(G_1)\cup V(G_2), (E(G_1)\...
- 203
2
votes
1
answer
117
views
Size of the orbit of a dense set
This question is a follow-up to: this post.
Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big ...
- 63
2
votes
1
answer
82
views
Structure of extensions arising in Lie approximation of connected groups
My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...
- 25.8k
2
votes
1
answer
206
views
Quasi-compactification of locally spectral spaces
Let $X$ be a locally spectral topological space (i.e. a space admitting an open cover by spectral spaces). Does there necessarily exist a quasi-compact locally spectral space $Y$ and an injective ...
user138661
2
votes
1
answer
134
views
How to determine the family of bounded functions from an infinite Fort space to $[0,1]$?
Definition: Let $X$ be a topological space and $b\in X$. We call $X$ a Fort space (with particular point $b$), when $X$ has topology $\{A\subseteq X: b \not\in A \; \text{or} \; X\setminus A\; \text{...
- 161
2
votes
1
answer
195
views
A question about semigroup union
The semigroup of all order-preserving and decreasing transformations in full transformations semigroup $T_n$ is denoted $C_n$.
I consider the idempotent set $A=\{\begin{bmatrix}2\\1 \end{bmatrix},\...
- 355
2
votes
1
answer
249
views
Scattered separators in Erdős space
Let $X$ be the set of all points in $\ell^2$ with all rational coordinates. $X$ is known to be totally disconnected, but $X$ is not zero-dimensional. For instance, the empty set does not separate the ...
- 3,317
2
votes
1
answer
159
views
First countable geometric realization of a simplicial group
Suppose we have a simplicial group $G$.
What do we need from $G$ to get first countable $BG$, where $BG$ is a geometric realization of $G$?
- 403
2
votes
1
answer
147
views
Space which is $T_1$ and sober but not Hausdorff?
Every Hausdorff space is $T_1$ and sober. Does the converse hold? I expect not. What's a counterexample?
I expected I should be able to look this up in Counterexamples in Topology, but unfortunately ...
- 64k
2
votes
1
answer
352
views
The completeness of spaces of continuous functions with the compact-open topology
For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology.
Problem. Is the space $C_k(X)$ Polish if it is Polishable ...
- 42k
2
votes
1
answer
324
views
Direct proof a property of hyperstonean spaces
First, let me state some basic facts and definitions for my question. I believe these are well-known among experts working on von Neumann algebras, but let me state them anyway since my question is ...
- 1,586
2
votes
1
answer
266
views
Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps?
I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...
- 85
2
votes
2
answers
134
views
On a generating set of numerical semigroups of multiplicity three
Let $S$ be a numerical semigroup. Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the Frobenius number of $S$; let $g(S)=|\mathbb ...
user111492
2
votes
1
answer
437
views
Covering properties of the rational sequence topology
This question is basically a rerun of this one, but I am curious about its resolution and couldn't really find info online on it, and no answers were given at the other site.
The rational sequence ...
- 5,407
2
votes
1
answer
122
views
If $H$ is essentially equimorphic to $K$, then is $H$ atomic only if so is $K$?
I will first state my question, and then give all the relevant definitions.
Q. Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic only if so ...
- 10.5k
2
votes
1
answer
368
views
Finding index/period of a semigroup element
The index and period of a finite monogenic semigroup $\langle x\rangle$ are the smallest numbers $i$ and $p$, respectively, satisfying $x^{i+p}=x^p$. The question is:
Is there an algorithm to find ...
- 1,651
2
votes
1
answer
211
views
Terminology for a monoid $(H, \cdot)$ s.t. $ax=a$ or $xa =a$ only if $x$ is a unit
Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied?
$$\text{(P) If }\,xy = x\...
- 10.5k
2
votes
1
answer
654
views
Triangulation induces regular CW complex structure
If a topological set is triangulable, dose the triangulation map gives it the (regular) CW complex structure? From definitions, I see it seems to be, but I am not that sure, for may exist some strange ...
- 103
2
votes
1
answer
266
views
What lattices are isomorphic to $R^{N}$ for some $N$, equipped with the product order?
What lattices are isomorphic to $\mathbb{R}^{N}$ for some $N\in \mathbb{N}$, equipped with the canonical order?
Remark:
When I say $\mathbb{R}^N$, I don’t mean it to be a vector space. Instead, I ...
- 97
2
votes
1
answer
248
views
a characterisation of proper maps via ultrafilters
Let $B$ be a topological space.
Call a subset $A\subset B$ ultrafilter-like iff $A$ is dense in $B$ and
each decomposition $A=A_1\cup A_2$,
into the union of two open subsets extends to a ...
- 113
2
votes
1
answer
266
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
2
votes
1
answer
364
views
Applications of topology to discrete dynamical systems?
I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets.
I mean cases where adding a topology to the sets ...
- 803
2
votes
1
answer
117
views
The separated uniform space associated with $(X,\mathfrak{U})$
If $\mathfrak{U}$ is a not necessarily separated uniform structure for some set $X$, then an equivalence relation $R$ can be introduced on $X$ by letting $x R y$ provided $(x,y)\in U$ for every $U\in \...
- 153
2
votes
1
answer
294
views
Finitely generated ordered monoids and noetherian subsets
(This question was asked a long time ago on MSE but got no answer so far...)
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...
- 6,700
2
votes
1
answer
179
views
Are all minimal totally separated spaces compact?
Let us call a space $(X,\tau)$ totally separated if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\neq \...
- 48.9k
2
votes
1
answer
223
views
pseudovarieties and profinite group : do * and g() commute?
Let $V$ and $W$ be pseudovarieties of finite monoids. We denote with $gV$ the pseudovariety of categories generated by $V$, and by $V*W$ the semidirect product of pseudovarieties $V$ and $W$.
Does ...
- 421
2
votes
1
answer
288
views
Snake-like continua and universal images
Among the Hausdorff compact spaces the closed interval is the simplest snake-like continuum. I'll present the definition after stating the problem.
The snake-like continua $\ S\ $ are universal ...
- 7,500
2
votes
1
answer
89
views
Constructivity of zeros demanded by topological degree
Let $f : S^{n - 1} \to S^{n - 1}$ be a smooth map from the unit vectors of $\mathbb{R}^n$ to themselves. If $f$ has nonzero degree, then we know that any smooth map $g : D^n \to \mathbb{R}^n$ ...
- 5,827
2
votes
1
answer
800
views
A question about Skorokhod metric
I have a question related to the Skorokhod distance.
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...
- 1,835
2
votes
1
answer
135
views
Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g.
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
Now define ...
- 1,835
2
votes
1
answer
128
views
Characterization of a subset of [0,1] $III$
I have a question related to the previous one.
Characterization of a subset of [0,1] $II$
Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e.
$t_n$ is said to converge to $...
- 1,835
2
votes
1
answer
145
views
Going Back-and-Forth Between Different Expressions/"Representations" for Open Books.
I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" ...
- 23
2
votes
1
answer
187
views
Unitization via "End points compactification"
We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{*}$ algebras. For example the one point compactification corresponds to the minimal ...
- 366
2
votes
1
answer
269
views
Homotopy with non piece-wise linear boundary
in the middle of a long proof I encounter the following problem.
Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand $E$...
- 21
2
votes
1
answer
433
views
Connectedness properties of groups of homeomorphisms
Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological ...
- 203
2
votes
1
answer
132
views
Maximal sub-inverse semigroups of $M_n(\mathbb{C})$ and $M_n(F_p)$
An inverse semigroup $S$ is a semigroup in which every element $x \in S$ has a unique inverse $y \in S$ such that $x = xyx$ and $y = yxy$. Are there some references characterizing the maximal sub-...
- 6,211
2
votes
1
answer
245
views
Probability measures on $L^p$
Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
- 8,512
2
votes
1
answer
291
views
Idempotents in Green J classes
I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular:
A $\mathcal J$-class containing an idempotent is called regular. ...
- 233
2
votes
1
answer
384
views
Properties of the weak-$*$ topology
Let $X$ be a topological affine space over a complete base field $\mathbb S := \mathbb C$, $\mathbb R$ or $\mathbb Q_p$. Let $X^*$ be the dual space of continuous affine functionals equipped with the ...
2
votes
1
answer
409
views
Extend Homeomorphism to Uniformly Continuous Function
I have a space $A$ which is homeomorphic to the open $n$-ball $B_n$.
I'm trying to build a CW-complex with it, so
I want a continuous function from the closed ball $\overline{B}_n$
to the closure $\...
2
votes
1
answer
796
views
Commutative, idempotent partially ordered monoids
A unital quantale is a suplattice with a compatible monoid structure. A quantale is called idempotent if it is idempotent as a monoid (every element is idempotent) (analogously for commutativity). ...
- 2,442