All Questions
825 questions
6
votes
1
answer
223
views
Minimal Hausdorff topologies compatible with a bunch of functions
Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...
- 48.9k
5
votes
1
answer
419
views
When is there an unbounded tower in $[\mathbb{N}]^\infty$?
(Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.)
This question assumes familiarity with combinatorial cardinal ...
- 3,104
5
votes
1
answer
269
views
A question on semi-stratifiable space
This question is also posted here.
A space $X$ is callled semi-stratifiable space if it has a $g$-function such that: for any point $x$ of $X$ and a sequence $\{x_n\}$ of $X$ if $x \in g(n,x_n)$, ...
- 654
5
votes
2
answers
231
views
Squaring a space with the fixed-point property
We say that a space $(X,\tau)$ has the fixed point property (FPP) if for every continuous map $f:X\to X$ there is $x\in X$ with $f(x) = x$.
What is an example of a space $X$ with FPP such that $X^2$ (...
- 48.9k
5
votes
1
answer
600
views
When is the generalized Cantor space $\kappa$-compact?
My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.
The generalized Cantor space is the space $2^\kappa$, with basic open ...
- 3,104
5
votes
1
answer
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 $...
- 11.8k
5
votes
3
answers
730
views
Is it possible to connect every compact set?
Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set.
Is there a always a compact connected $L\subset X$ such that $K\subset L$?
This ...
- 5,529
5
votes
0
answers
263
views
Are continuous self-maps of the Golomb space $\mathbb G$ dense in the space of all self-maps of $\mathbb G$?
The Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $a,b$ ...
- 41.9k
5
votes
0
answers
228
views
What is the smallest number of hyperplanes covering $\ell_2$?
For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$.
By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
- 41.9k
5
votes
1
answer
287
views
Is each compactification of $\mathbb N$ soft?
Definition. A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be soft if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\...
- 41.9k
5
votes
2
answers
454
views
Is each locally compact group topology on the permutation group discrete?
Question. Is each locally compact group topology on the permutation group $S_\omega$ discrete?
Here $S_\omega$ is the group of all bijections of the countable ordinal $\omega$. A group topology on a ...
- 41.9k
4
votes
1
answer
364
views
Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$
Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread.
Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
- 10.5k
4
votes
1
answer
1k
views
Quotients of standard Borel spaces
Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\...
- 1,655
4
votes
1
answer
668
views
special extremally disconnected spaces with only finite isolated points
We Know that a cardinal $\kappa$ is measurable if there is a set $X$ with cardinal $\kappa$ and a {0,1}-measure $\mu: P(X) \rightarrow ${$0,1$} so that for all $x \in X$, $\mu(x)=0$ and $\mu(X)=1$. ...
- 1,788
4
votes
2
answers
558
views
Is a specific sequentially closed subset of $M([0,1])$ closed?
Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
- 95
4
votes
0
answers
114
views
Find at least one square-boxed subcontinuum
Recall that a plane continuum is a closed, bounded,
connected subset of the plane.
It is non-degenerate if it contains at least two points.
(We may sometimes just say "continuum" even if
we ...
- 1,375
4
votes
1
answer
169
views
Is every invertible-free cancellative monoid action represented by "shifting" certain maps?
[Note: This question is closed. It's current content reflects a draft of a potential new question, modified from the original by adding conditions to the premises; see comments]
Let $W,X$ be ...
4
votes
0
answers
111
views
Does an interior point necessarily pass through the boundary under a homotopy?
It's a straightforward exercise to show that if a point moves continuously from the inside of a set to the outside, it necessarily passes through the topological boundary of the set. This question is ...
- 391
3
votes
1
answer
395
views
Closed embedding into a normal Hausdorff space and left lifting property
I am trying to understand the characterization of the class of closed embeddings into a normal Hausdorff space as the class of continuous maps satisfying the left lifting property with respect to a ...
- 3,901
3
votes
1
answer
203
views
Centralizer of a single element in the monoid of self-maps of a set
This is a follow-up to this question: For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?
Let $X$ be a set, and $X^...
- 63.9k
3
votes
4
answers
934
views
Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?
Q1.
Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $...
- 1,375
3
votes
0
answers
78
views
Nowhere dense covering number of a connected $T_2$ space
This is a generalization of an older question.
If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a ...
- 48.9k
3
votes
1
answer
122
views
A BF-monoid $H$ s.t. $H^\times$ is not divisor-closed
Let $H$ be a (multiplicative) monoid, and denote by $H^\times$ the set of units of $H$ and by $\mathcal A(H)$ the set of atoms of $H$ (let me recall that an element $a \in H$ is an atom if (i) $a \...
- 10.5k
3
votes
1
answer
440
views
In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?
In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
Note that ...
- 4,862
3
votes
2
answers
165
views
Weak ideal systems $r$ for which the $r$-coheight satisfies a kind of triangle inequality
Let $H$ be a multiplicatively written, commutative monoid with identity $1_H$, and let $\mathcal P(H)$ be the power set of $H$. If $X, Y \subseteq H$, we will set $$XY := \{xy: x \in X,\, y \in Y\}.$$
...
- 10.5k
3
votes
1
answer
228
views
Computing the Heyting operation on the frame of nuclei
(The following definitions are meant to be standard and are reproduced for completeness of the question.) A frame is a partially ordered set in which every finite subset has a greatest lower bound (“...
- 32.5k
3
votes
1
answer
149
views
Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods
Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?
- 1,145
3
votes
1
answer
146
views
Maximal elements in the partially ordered set of image spaces
If $(X,\tau)$ is a topological space, let $\text{Im}(X)$ denote the collection of subsets $S$ of $X$ such that there is a continuous function $f:X\to X$ with $\text{im}(f) = S$.
Is there a space $(X,\...
- 48.9k
2
votes
1
answer
301
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
0
answers
82
views
Enveloping a Jordan curve with a trace of another one
This question is inspired by this one, or rather the way I understood it.
Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\...
- 5,529
2
votes
2
answers
447
views
Reconciling some result about the exponential map, the Chow-Rashevskii theorem, and $\mathrm{Diff}_0(M)$
Let $M$ be a $C^{\infty}$ manifold $C^{\infty}$-diffeomorphic to $\mathbb{R}^d$. I've recently come across some results which I'm trying to reconcile. Let $\mathfrak{X}(M)$ denote the set of ...
- 5,405
2
votes
0
answers
406
views
Complete topological groups in which all subgroups are closed
My previous question has been answered by YCor; so I am asking a new one with a reasonable additional assumption. See the previous question for the background and motivation.
General question: does ...
- 15.6k
2
votes
0
answers
159
views
Are there hereditarily square-boxed plane continua?
A plane continuum is a bounded, closed and connected subset of the plane.
A bounding box $B$ for a plane continuum $C$ is
a rectangle $B=[a,b]\times[c,d]$ (including sides and interior)
such that $C$ ...
- 1,375
2
votes
1
answer
142
views
Is every semi-stratifiable space $\omega$-monolithic?
Is every semi-stratifiable space $\omega$-monolithic?
Definitions
A topological space $(X,\tau)$ is called semi-stratifiable if there exists a function $g:\omega\times X\to\tau$ such that:
for any ...
- 621
2
votes
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
2
votes
1
answer
382
views
Continuous real function on germs
Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C_0^{m,n}$...
- 21.5k
2
votes
1
answer
200
views
Are the connected components of a Priestley space closed?
Preliminaries A Priestley space is both a poset and a topological space. The topologically connected components of the space are trivially closed. (They are just the points of the underlying set.) But ...
- 2,464
2
votes
1
answer
183
views
Maximal connected topologies
We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected.
If $(X,\tau)$ is ...
- 48.9k
2
votes
4
answers
535
views
Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$
Let $(X,\tau)$ be a topological space. Let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$.
What can be said about spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$? For ...
- 48.9k
1
vote
1
answer
142
views
Congruences that aren't "finite from above," take 2: semigroups
This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...
- 20.5k
1
vote
0
answers
256
views
partially commutative monoid [closed]
Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
- 1,269
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
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
1
vote
1
answer
130
views
Two consecutive continua
Are there two non homeomorphic continua $X,Y$ such that $X $ can be embedded in $Y$ but there is no topological space $Z$ with $$X<Z<Y.$$
The later relation means that $Z$ ...
- 356
1
vote
1
answer
388
views
About isotopy of simple close curve
In the Primer mapping class group by farb Margalit. We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if ...
- 119
1
vote
1
answer
206
views
Totally non fixed point property
Edit: According to the comment of Pietro Majer, I revise the question
Is there a non singleton compact connected Hausdorff topological space $X$ for which the following property hold?:
"Constant ...
- 356
1
vote
3
answers
345
views
Under what general conditions is the set $S := \left\{\int_{X}v(x)\pi(x)\,\mathrm{d}P(x) \mid \pi: X \to A\right\}$ closed?
Let $X$ be a compact subset of $\mathbb R^n$ and let $A$ be a compact subset of $\mathbb R^k$. Let $P$ be a probability distribution on $X$ and $v$ be a $P$-measurable function from $X$ to $\mathbb R^{...
- 6,853
1
vote
1
answer
160
views
Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimensional Euclidean spaces
For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple ...
- 6,215
0
votes
0
answers
81
views
Let $S$ be a surface, $K$ compact in $S$ with finitely many components. Does the frontier of a component of $S-K$ have finitely many components?
Let $S$ be a connected surface and $K$ a compact subset of $S$ with finitely many connected components. Let $U$ be a connected component of $S-K$. Does the frontier of $U$ in $S$ have finitely many ...
0
votes
1
answer
281
views
Does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and …?
Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with non-zero finite length $L$. Then, does there always exist a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
- 113