All Questions
5,184 questions
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635
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Do homotopic non-intersecting simple closed curves separate the surface?
Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
They are homotopic to each other.
They are set-theoretically disjoint.
Is $S\setminus(C_1 \cup C_2)$ ...
user13946
2
votes
0
answers
371
views
Descriptive set theory on $\mathbb{R}^\mathbb{N}$
The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...
- 20.5k
-1
votes
1
answer
110
views
Variety of commutative semi group [closed]
V is a variety of commutative semi group satisfying the identity $x^2 = x^3$.
I need to prove that:
$|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$.
Any hints on this ?
$F_V$ is V-free algebra.
- 155
2
votes
2
answers
439
views
countably complete filters
Is there any description of the set of countably complete filters on the lattice of dense $G_{\delta}$ subsets of a compact, second countable metric space? [I haven't just dreamt this up: it describes ...
- 607
1
vote
0
answers
103
views
Lower periodic subsets of groups and semigroups
Suppose that $A$ and $B$ are subsets of a group or semigroup. We call $A$ left
upper [resp. lower] $B$-periodic if $BA\subseteq A$
[resp. $A\subseteq BA$]. If $A$ is both left upper and
lower $B$-...
- 995
3
votes
1
answer
376
views
Chaos in uniform spaces
Let $Dom$ be a uniform space, and $\hspace{.04 in}f$ be a continuous function from $Dom$ to itself satisfying:
For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural
number $n$ ...
user5810
1
vote
1
answer
172
views
Normal Uniform Spaces and their function uniform spaces
Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase
$$\Lambda =\{ \{(f,...
user31967
0
votes
1
answer
231
views
A question on linearly lindelof space
Let $X$ is a linearly lindelof subspace of $Z$ and $b$ is not $\omega$-separated from $X$, i.e., for any closed $G_\delta$ set $P$ of $Z$ which contains $b$, $P\cap X \not=\emptyset$. If $\tau < \...
- 654
5
votes
1
answer
304
views
flat maps of monoids which are not localizations
It is well known that a localization $S^{-1}R$ of a commutative ring $R$ is flat as a $R$-module.
Rather, I am looking for extensions of rings which share certain properties of localizations, like ...
- 6,210
1
vote
0
answers
130
views
Regarding graphs of continuous functions between zero dimensional spaces
Background for the question: Let for any topological space $B$, $I(B)$ denote the topological space which has the same set of points as of $B$, and the topology is generated by closed and open sets of ...
1
vote
1
answer
118
views
Banach Isomomorphic Cts Fucntion Algebras for two Non-Homeomorphic Top Spaces?
Can anyone provide me with an example of two non-homeomorphic locally-compact Hausdorff spaces $X$ and $Y$, such that $C(X)$ and $C(Y)$ are isomorphic as Banach algebras. Clearly, the Gelfand--...
- 1,453
9
votes
0
answers
741
views
Pseudocycle definition of open Gromov-Witten invariants
I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as long!...
- 1,129
2
votes
1
answer
274
views
Does X have any diagonal properties?
Assume that $2^{\omega_1}=2^\omega=\mathfrak{c}$. Let $D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$={$\xi<\mathfrak{c}:y(\xi)=1$}, the support of $y$, and let ...
- 654
3
votes
0
answers
103
views
Paracompact and countably compactly generated spaces
A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces.
Are countably compactly generated spaces paracompact spaces? Do we have ...
- 31
3
votes
1
answer
206
views
Topological question about right-lifting property and the evaluation map
Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ ...
- 3,901
15
votes
0
answers
2k
views
Covers of $Z^k$
This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...
1
vote
1
answer
1k
views
A question about unbounded connected subsets of the plane.
A number of clever examples have been given of unbounded connected subsets of the
Euclidean plane containing no infinite bounded subsets that are connected. None of those that
I have seen are ...
- 6,215
6
votes
3
answers
372
views
Notion of finite dimensional simplicial space
I was wondering, what a $N$-dimensional simplicial space $X$ should be. Of course the degeneracy maps force the spaces to be nonempty in high dimensions. Currently I have two different versions and i ...
- 11.1k
9
votes
1
answer
1k
views
When completion of locally compact length space is locally compact?
As far as I know the answer to the question:
"Is it true that a completion of a locally compact length space is locally compact?" - Negative.
Does anybody know some metric and/or topological ...
- 141
1
vote
2
answers
686
views
Existence of convergent subsequences for all values in range?
Consider sequence $s(n) = \sin{nx}$. Are there values of $x$ for which the following holds: For every $y \in \[-1,1\]$ there is a subsequence of $s(n)$ converging to $y$? (Or perhaps just for the open ...
- 367
1
vote
1
answer
136
views
Nonhomeomorphic CW-complexes that are "stably" homeomorphic
Do there exist CW-complexes $X$ and $Y$ that are not homeomorphic, but $X \times I$ and $Y \times I$ are homeomorphic? Here $I$ denotes the unit interval $[0, 1]$.
- 19
1
vote
0
answers
275
views
Regular Borel Measures equivalent definition
Please help me understand how the below definition is equivalent to the standard definition of regularity which says that a measure is regular if for which every measurable set can be approximated ...
- 11
0
votes
1
answer
341
views
Length of intersection of intervals
Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
- 500
4
votes
3
answers
746
views
A question about totally disconnected point sets.
Let H be an infinite dimensional and separable Hilbert space. Let C be a closed and
connected subset of H containing more than one point. Can C ever be the countable union
of closed and totally ...
- 6,215
0
votes
1
answer
224
views
Special functions on the unit disk
Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.
We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a ...
- 1,271
1
vote
0
answers
167
views
How many ways we have to prove that a topologically (or analytically) nice mapping is injective?
I would like to know what are the methods people have used to prove that a topologically (or analytically) nice mapping $f: B\to \Omega$ is injective? Above, $B$ is the unit ball in $\Bbb R^n$ and $\...
- 1,881
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-...
- 6,201
6
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0
answers
189
views
Pettis Integrability and Laws of Large Numbers
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
- 8,512
1
vote
1
answer
163
views
Term for number of crossings of smooth curves
Two smooth oriented finite curves $g_1, g_2$ on e.g. the 2-dimensional torus can intersect each other transversally in two ways: either the pair $(Tg_1(x),Tg_2(x))$ of tangent vectors in the ...
3
votes
2
answers
699
views
Conditions useful for proving paracompactness
I have a family of properties which I want to show taken together imply paracompactness (I can show that they are all implied by paracompactness). I can prove a whole bunch of things which are ...
- 1,331
2
votes
1
answer
232
views
Could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology?
Let $X$ be a Tychonof space and $\beta X$ is its compactification. Then could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology?
- 654
0
votes
3
answers
404
views
Some Questions about zero-dimensional subsets of the unit interval related to cantor set
Let $\mathbb{P}$ denote the set of all irrational numbers in the open segment$(0 , 1)$. let $K$ be the intersection of $\mathbb{P}$ and the standard cantor set and $H=\mathbb{P}-K$. as you know these ...
- 1,788
1
vote
2
answers
341
views
A question about connectedness in Euclidean space [closed]
Here is a question which seems true to me but I can't rigorously show. Suppose $K$ is a compact subset of $\mathbb{R}^n$ such that $\mathbb{R}^n\setminus K$ is connected, does it follow that for any ...
- 23
1
vote
0
answers
264
views
Z-sets in the Hilbert cube
If $(X,d)$ is a metric space, then we say that a closed subset $A$ of $X$ is a z-set if for each number $k\gt 0$ there is a continuous map $f_k$ from $X$ into $X-A$ such that $d(x,f_k(x))\lt k$.
I ...
- 11
2
votes
0
answers
248
views
A question about connected subsets of metric spaces
Let M be a metric space. Let T(M) be the topology of M (i.e. the collection of all open subsets of M)
and let C(M) be the collection of all connected subsets of M. In my opinion one often has a much
...
- 6,215
1
vote
0
answers
321
views
Type I subspaces of the Stone Cech compactification of $\omega$
EDIT: I found a construction, see below. I decided not to delete the question in case someone is interested.
A space $X$ is of Type I if $X=\cup_{\alpha<\omega_1} X_\alpha$, where each $X_\alpha$ ...
- 1,855
7
votes
0
answers
517
views
Is there a natural topology on the set of open sets ?
Given a topological space $(X,\mathcal{O})$ can one assign a natural topology to $\mathcal{O}$ such that
1) The intersection of a compact set of open sets is again open,
2) The maps $\cap,\cup:\...
- 11.1k
0
votes
1
answer
151
views
Is there any result concerning on the metric dimension of inverse limit?
To be specific, my question is as follows:
Question: Let $X$ be an inverse limit of compact metric spaces $(X_i, d_i)$, then does it hold
$\dim(X, d) \leq \sup_i \{\dim (X_i, d_i)\}$ for some ...
2
votes
0
answers
144
views
Hall's paper on the profinite groups and Andre Weils "voisinage" notion
I am reading through a classical paper A Topology for Free Groups and Related Groups
by Marshall Hall Jr. in which profinite groups are defined for the first time.
There he defines on p. 129:
...
- 798
3
votes
1
answer
292
views
Can a closed trefoil appear as a space-time "cut" of an open trefoil?
An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space, in space-time is a knotted hypersurface.
Different observers in space-time have ...
- 75
4
votes
1
answer
243
views
When is Prim(A) of an infinite discrete group hausdorff ?
Does anyone know, if the following result has been proved ?
Let G be an infinite discrete group. A = L1(G) it's algebra and Prim(A) the set of prime ideals with spectral topology.
The result is :
...
- 41
0
votes
1
answer
134
views
Two different products of filters
By filters I will mean filters on some set $\mho$.
I define product of an infinite family of filters in two ways. I feel (by analogy with properties of Tychonoff product vs box product of topological ...
- 765
5
votes
2
answers
482
views
Is a map that is locally fiberwise equivalent to a product a Hurewicz fibration?
The following is a result I feel like I've seen some form of before, but can't figure out how to prove or find a reference for. Suppose you have a map p:E \to B, with B paracompact, and suppose that ...
- 31.2k
1
vote
1
answer
515
views
Braid*Temperley-Lieb=?
I would be very astonished if this algebra isn't named.
You simply have the braid AND the Temperley-Lieb generator in
the algebra. Rules are the usual Reidemeister equivalents
plus the kink and ...
- 4,829
2
votes
1
answer
512
views
Question about analytic curves
Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true ...
- 203
3
votes
0
answers
877
views
The "pullback presheaf" and the proper base change theorem in topology
Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$
be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$:
$$
V\...
- 7,609
3
votes
1
answer
239
views
Function spaces over pseudocompact spaces
Let $K$ be a pseudocompact Tychonoff space that is, a Hausdorff $T_{3.5}$ space for which every continuous function $\varphi \colon K\to \mathbb{C}$ is bounded. Let $\beta$ be the Stone-Cech extension ...
- 41
9
votes
0
answers
685
views
Name for a topological space where every closed set contains a closed point
A coauthor and I have stumbled upon a useful topological property -- namely, we are interested in the property that every nonempty closed set contains a closed point. However, neither of us are ...
- 1,802
7
votes
0
answers
624
views
"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
- 18.7k
0
votes
2
answers
359
views
some questions on Lindelöf property
I have several questions on Lindelöf property.
If every point countable open cover of $X$ has a countable subcover (Condition A), does $X$ have Lindelöf property? How far is having Condition A from ...
- 654