All Questions
5,184 questions
4
votes
1
answer
660
views
A question about Moore spaces.
Moore in his classic book "Foundations of Point Set Theory" defines a "continuum" to be a set of points
that is both closed and connected (but not necessarily compact). He also calls "non-degenerate" ...
- 6,215
6
votes
0
answers
366
views
Whitney approximation without second countable
One version of Whitney's approximation theorem states the following:
Let $N$ be a smooth, Hausdorff, second-countable (or paracompact) manifold, then given any continuous function $F:N\to \mathbb{R}...
- 39.1k
1
vote
0
answers
51
views
Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR
Let $D^n$ be the closed unit ball in $\mathbb{R}^n$. Given a compact, $n$-dimensional, AR(Absolute Retract) metric space $X$, must it happen that either $X$ embeds in $D^n$ or $D^n$ embeds in $X$?
...
- 11
1
vote
0
answers
40
views
Decomposition which is locally connected
It is possible construct a connected compact metric space $X$ and a continuous decomposition $\mathcal{G}$ of $X$ that satisfies:
1)$X/\mathcal{G}$ is locally connected.
2)If $M$ is a compact ...
- 11
-1
votes
2
answers
466
views
Union of proximally connected sets
Let (δ;U) is a proximity space.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
Is the following true? (I need a proof or a counter-example.)
Conjecture If S ...
- 765
1
vote
2
answers
504
views
Do all graphs of C1 functions have Hausdorff dimension 1?
Suppose f is a real-valued function of one variable, and suppose f is of differentiability class C1. My question is, if $\Gamma$ is the graph of f, then must $\dim_H(\Gamma)=1$? If anyone knows of a ...
2
votes
0
answers
138
views
Topology of Asymmetric Symmetric Products
Let $X_1,...,X_m$ be connected, simply-connected CW sub-complexes of a CW complex $X$. Let the symmetric group on $m$ letters, $S_m$, act on $P:=X_1\times\cdots\times X_m$ in $X^m$ by permuting ...
- 8,529
0
votes
0
answers
148
views
Only finitely many fundamental groups in $M(n,k,v,D)$?
Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...
- 689
4
votes
1
answer
675
views
Name for topology making group action continuous
Fix a set $X$ with right $G$-action. Give $X$ a topology $\tau$ and make $G$ a topological group. (These topologies need not make the action continuous).
We can define another topology $\tau'$ on $...
- 2,895
0
votes
0
answers
128
views
minimal (strongly) KC
If P is a topological property, then a space (X, τ) is said to be minimal P (respectively, maximal) if (X,τ) has property P but no topology on X which is strictly smaller (respectively, strictly ...
- 21
1
vote
1
answer
374
views
Weak convergence of measures on non-metrizable spaces
(ZF + Countable Choice)
Let $\langle X,\mathcal{T} \hspace{.06 in} \rangle$ be a second-countable Hausdorff space. Let $\mu$ be a Borel measure on $X$.
Let $\langle I,\leq_I \rangle$ be a directed ...
user5810
1
vote
1
answer
101
views
ball in universal cover belongs to the union of actions on a section?
M is an n-dim manifold. $\pi :\tilde M \to M$ the universal cover of M. $\tilde p \in \tilde M$ a lift of p. We choose a measurable section $j:{B_1}\left( p \right) \to {B_1}\left( {\tilde p} \right)$,...
- 689
2
votes
1
answer
220
views
homeomorphisms on k-spaces
Let X be a Hausdorff k-space (Hausdorff compactly generated space) and h a bijection on X such that for any subset E of X we have
...
- 23
5
votes
1
answer
293
views
semigroups acting as continuous functions on regular rooted trees
Let $T$ be a regular rooted tree. Make $T$ into a metric space by making each edge isometric to the unit interval. What is known about what semigroups can act as continuous functions on $T$ such ...
- 51
1
vote
0
answers
65
views
The topology of complete minimal surfaces of finite total Gaussian curvature [closed]
Suppose that M is
a complete minimal surface with finite total curvature. If M is embedded in $\mathbf{R}^3$, then
we observe that M viewed from infinity looks like a plane passing through the ...
- 11
2
votes
1
answer
214
views
union of Stone-Cech remainders
Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
- 607
4
votes
0
answers
331
views
What is the pro-algebraic completion of the free semigroup on one generator?
This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view.
Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...
- 38.6k
1
vote
0
answers
145
views
Reference needed: Does pseudo laminated compact subsets of the plane separate the plane?
Hi,
doing my research I found the following problem and I´ll be glad if someone could give a reference.
We say that a compact connected subset $K$ of the plane is psuedo laminated if the following ...
- 19
3
votes
0
answers
166
views
A question of terminology - Unitizations of semigroups
There are at least two standard ways of unitizing a (small) semigroup $\mathbb A$:
(i) We add an identity regardless that $\mathbb A$ is already unital.
(ii) We add an identity only if none is ...
- 10.5k
6
votes
0
answers
108
views
How to call a point in a space having the property that there is essentially one $\omega$-sequence converging to it?
Consider the point $x=\langle \omega_1,\omega\rangle$ in the Tychonov plank $(\omega_1 + 1)\times(\omega + 1)$. Then there is essentially only one sequence (of length $\omega$) converging to it, ...
- 1,855
1
vote
1
answer
492
views
Сomplete homogeneous space which is not locally compact
It is well-known theorem that every locally compact, homogeneous, metric space is complete.
Does anybody know example of complete, homogeneous, metric space which is not locally compact?
- 141
2
votes
0
answers
61
views
Removable sets for simply connectedness of a differentiable manifold
I am sorry that my question might be stupid for experts, but I really do not know the answer.
Let $M$ be a smooth $n$-manifold. We assume that there exists a distance $d$ on $M$ such that $(M,d)$ is ...
- 1,881
8
votes
1
answer
655
views
Coherent spaces
In Proofs and Types, Girard discusses coherent (or coherence) spaces, which is defined as a set family which is closed downward ($a\in A,b\subseteq a\Rightarrow b\in A$), and binary complete (If $M\...
- 475
5
votes
0
answers
115
views
Equivariant zero dimensional extension recovering a given measure
Let $X$ be a compact metrizable space and $\alpha: \mathbb{Z}^d\curvearrowright X$ a continuous group action. Then it is well known that there exists a zero dimensional compact space $Y$, an action $\...
- 1,023
2
votes
0
answers
131
views
Topological dimension of quotient group determined by the inverse limit of discrete free monoids
Must the natural quotient group of the inverse limit of a sequence of nested discrete free monoids have topological dimension zero?
The question might well be open, but I would be grateful for news ...
- 1,968
1
vote
0
answers
130
views
A question on star $\sigma$-compact spaces
The question is also posted here.
A topological space $X$ is said to be star $\sigma$-compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a $\sigma$-compact subspace $K$ of $X$ such ...
- 654
2
votes
0
answers
126
views
A question on continuous mappings
The question is also posted here.
Let $M=\mathbb{R}$ and $\tau_M=\lbrace U\cup A: U$ open in $\mathbb{R}, A\subset \mathbb{R} \setminus B\rbrace$, where $B$ is a Bernstein set. Then $(M,\tau_M)$ is a ...
- 654
3
votes
0
answers
61
views
Local cross sections in infinite dimensional groups
Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples ...
3
votes
1
answer
148
views
Metric on the set of Polyhedral Decompositions of a Compact Metric Space
Let $(X,d)$ be a compact metric space of finite diameter. Recall that we can always compute the Hausdorff distance between subsets $A, B \subset X$ via
$$ d_H(A,B) = \max\left[\sup_{a\in A}\inf_{b\in ...
- 15.5k
3
votes
0
answers
107
views
Hindman's theorem variant for noncommutative semigroups
The well set proof of Hindman's finite sums theorem applied to noncommutative semigroups yields a sequence of elements such that finite products ordered coherently with this sequence are in one set. ...
- 31
0
votes
0
answers
196
views
measurable function on a locally compact space for a regular measure
A well known classical fact is that a Lebesgue measurable function on Euclidean space is almost everywhere equal to a Baire class 2 function. A relatively modern reference for this fact is van Rooij -...
- 1,704
2
votes
1
answer
218
views
Shrinkable maps and universal weak equivalences
Recall that a morphism $f:X \to B$ is called shrinkable is there exists a section $s:B \to X$ together with a homotopy $$H:I \times X \to X$$ from $sf$ to $id_X$ over $B,$ i.e. for all $t$, the map $$...
- 15.5k
0
votes
1
answer
209
views
On generalized ordered spaces
Let X be a Go space. If G is open in X, why is every convex component of G open?
( It is well known that any non-void subset G of X can be uniquely represented as a union
of its maximal convex ...
- 654
1
vote
0
answers
83
views
Topologies on spaces of linear sections
Let $X$ and $Y$ be topological linear spaces which are complete & Hausdorff, and admit dual spaces which separate points. Suppose the topologies are non-separable and non-metrizable.
Let $f : X \...
- 8,512
1
vote
1
answer
148
views
Staggered timing on 2-D random walks by multiple agents
In 2-D lattice random walks by multiple drunks who can't step onto each other, mathematically I would just say the whole cellular automaton updates "at once".
But to simulate this on a computer, I ...
- 507
4
votes
0
answers
158
views
Does this construction yield an injective hull ?
Let $K$ be an object of $\mathbf{CHaus}$, the category of compact Hausdorff spaces, and $K \xrightarrow{\ \ \sigma \ \ } K$ be an involutory morphism without fixed points. Define $C^{\sigma}(K)$ as ...
- 7,249
2
votes
0
answers
124
views
Reasoning about "approximately" associative structures and "almost monoids".
If $(M,+)$ is a monoid then it obeys the laws:
$$m_1 + 0 = 0 + m_1 = m_1$$
$$m_1+(m_2+m_3)=(m_1+m_2)+m_3$$
But what if I have a structure $(A,+)$ that is almost a monoid, but not quite. For example,...
- 21
4
votes
1
answer
240
views
Transversals to singular subvarieties
Say $\mathbb{C}^d \subset Y^{N-k} \subset \mathbb{C}^N$ are closed imbeddings of complex analytic subvarieties of the indicated dimension, $Y$ is not smooth. At a point $y \in Y$, a generic, ...
- 8,723
5
votes
0
answers
104
views
Is a closed set with orbit capacity zero automatically thin?
Let $G$ be a countably infinite amenable group. Let $\alpha: G\curvearrowright X$ be a continuous group action. (Mostly free and minimal, though!)
Definition 1: Let $A\subset X$ be closed and $U\...
- 1,023
2
votes
0
answers
73
views
A construction with Hyperspace of continums
Let $X$ be a compact connected metric space. Its hyperspace is denoted by $2^{X}.$ $X$ is considered as a subset of $2^{X}$ via the embedding $x\mapsto \{x\}$. Assume that $f:X\to X$ is a ...
- 356
0
votes
1
answer
285
views
A Nomenclature Issue : Imprimitive Semigroup?
The following question was asked by me on the forum sci.math.research,
“An imprimitive group is a transitive permutation group with a non-trivial
equivalence relation compatible with the action of ...
- 113
0
votes
2
answers
505
views
Partition into connected sets by proximity
Let (δ;U) is a proximity space.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
I will call connected component a maximal connected set.
Is this true: U is ...
- 765
2
votes
0
answers
192
views
Are open convex PL subsets of R^n PL homeomorphic to R^n?
This is a basic issue of PL topology that I assume must be true, but I can't find a written reference: is a convex open PL subset of $\mathbb R^n$ PL homeomorphic to $\mathbb R^n$? I've scanned ...
- 5,534
5
votes
0
answers
263
views
Coloring $\mathbb{Z}^k$ and a fixed point theorem
This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in ...
user6976
5
votes
0
answers
196
views
Is there a Whitney-type theorem Cauchy manifolds?
Let $M$ be a Cauchy space whose induced topological space is a second-countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^m$.
Does it follow that there exists a subspace $N$ of $\...
user5810
5
votes
0
answers
308
views
Properties of the Zariski-Riemann topology on the set of valuations
One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...
- 51
2
votes
1
answer
307
views
Name for a kind of topological property?
What should I call a property (P) of (open) subspaces of a space $X$ such that:
If $U$ satisfies (P), then so does every open subset $V\subset U$
If {$U_i$} is a pairwise disjoint collection of ...
- 12.5k
5
votes
1
answer
329
views
Example of a quasitopological group with discontinuous power map
A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion $G\...
- 7,193
0
votes
1
answer
94
views
What is the way to after finding Cohen-Macaulay semigroup as ring of monomial?
http://people.missouristate.edu/lesreid/reu/2007/PPT/robin.ppt
it said missing part inside and not missing part outside is non Cohen-Macaulay semigroup
which determine whether is missing in the most ...
- 1
0
votes
3
answers
248
views
how slow can the dimension of a product set grow?
Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:
$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,
where $\sim$ denotes "...
- 1,839