All Questions
5,185 questions
7
votes
2
answers
305
views
Functionals on oriented matroids
Oriented matroids are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors.
Let $R=\lbrace 0,+,-\rbrace$ with the monoid ...
- 38.6k
2
votes
1
answer
233
views
Continuous Functions
Is there a pair of continuous surjective functions say $f_1$ and $f_2$ from $\mathbb{Q}$ to itself such that for every $x, y \in \mathbb{Q}$, $f_1^{-1}(x) \cap f_2^{-1}(y)$ is non-empty?
- 175
8
votes
1
answer
657
views
What should the morphisms in the Category of Directed Sets be?
Directed sets are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ ...
- 352
5
votes
0
answers
912
views
Decreasing sequence of closed convex sets in a Banach space
Let $(C_n)$ be a decreasing sequence of closed convex subsets in a Banach space $(E,\Vert \cdot \Vert)$. The question I have is about the content of $C=\bigcap_{n=0}^\infty C_n$.
If the $C_n$ are ...
- 1,355
4
votes
1
answer
219
views
Is the following product-like space a Polish space?
Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-...
- 6,287
2
votes
0
answers
50
views
Nonautonomous wave equation of memory type
I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written ...
- 617
1
vote
1
answer
686
views
dual space of the quotient space of some locally convex topological space
I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the ...
- 1,835
3
votes
0
answers
182
views
LCH topologies on Groups that are not group topologies
Ellis's 1957 paper on Locally Compact Transformation groups proves the following:
A locally compact hausdorff topology on a group $(G, \cdot)$ for which left and right multiplication are (separately)...
- 352
2
votes
1
answer
141
views
Is the interval topology of $(\mathbb{N}^\mathbb{N}, \leq^*)$ connected?
Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq ...
- 48.9k
9
votes
1
answer
547
views
Constructible sets in Hausdorff spaces
In an attempt to write a proof by contradiction, I end up with a space $X$ with the following properties:
(0) $X$ is nonempty,
(1) $X$ is Hausdorff,
(2) $X$ has no isolated points,
(3) every subspace ...
- 19.6k
1
vote
1
answer
71
views
Every open convex-valued multimap has global sections?
Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is ...
- 1,368
1
vote
1
answer
330
views
Two questions on path connected spaces
Is it true to say that a compact hausdorff space $X$ is path connected if and only if for every continuous function $f:X\to \mathbb{C}$, we have $f(X)\subset \mathbb{C}$ is path connected?
2....
- 366
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
1
vote
1
answer
370
views
Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space
This is a follow-up to this question.
Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$).
I'm interested in the topological ...
- 191
6
votes
1
answer
338
views
Topological groups defined by completely disconnected subgroups
Can you define a group topology on a group by specifying which subgroups should be discrete with respect to that topology (where a subgroup $S$ of $G$ is discrete if each $s\in S$ has an open ...
- 345
8
votes
2
answers
2k
views
End point compactification for metric spaces
Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...
- 3,033
1
vote
2
answers
2k
views
General criteria for exhaustion by compact sets
Hello,
I'm wondering if there exists general results insuring exhaustion by compact sets of a given topological space ?
nicolas
- 113
3
votes
1
answer
107
views
Topology with no direct lower neighbor
Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete ...
- 48.9k
3
votes
0
answers
198
views
Properties of convergence at points of continuity
Let $J$ denote the set of functions $f : [0, \infty) \to \mathbb{R}$ that are right-continuous and have left-hand limits (r.c.l.l.) and such that their points of discontinuity are jumps.
Then $J$ is a ...
- 1,773
9
votes
0
answers
375
views
A compact T1 topological space has a proper dense subset to which it is homeomorphic. What can be said about the space?
Let $X$ be a compact T1 (so singleton subsets are closed) topological space. Suppose that there is a proper subset $D \subset X$ such that:
$D$ is dense in $X$;
$D$ is homeomorphic to $X$.
Note that ...
1
vote
3
answers
172
views
Is the poset of all precompact group topologies on an abelian group $G$, order-isomorphic to $\operatorname{Sub}(\hat{G})$?
In this page, in abstract, it is claimed that the poset of all Hausdorff precompact group topologies on an abelian group $G$, is order-isomorphic to the the subgroup lattice of $\hat{G}$, the ...
- 1,145
2
votes
0
answers
360
views
Given locally compact and $\sigma$-compact, can we get partition of unity?
Let $X$ be a locally compact, $\sigma$-compact Polish (complete and separable metric) space. How to prove: "There is an increasing sequence of continuous cut-off functions with compact support, $0\...
- 471
0
votes
1
answer
299
views
Can the image of a disk have nontrivial Hausdorff measure for $1 < d < 2$?
I am reading a blog which talks about a $C^1$ diffeomorphism $f: \mathbb{D}\{ x^2 + y^2 < 1\} \to \mathbb{R}^2$ and estimates the Hausdorff dimension of its image $\mathcal{H}_\sqrt{2}^d (f(\mathbb{...
- 22.8k
2
votes
0
answers
459
views
Weak topology on subsets of a normed space
I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset.
When is the norm a continuous function on $E$?
When is the metric induced by the ...
- 5,529
5
votes
1
answer
450
views
Connection between subnet and superfilter
Let's define a net and subnet in this way:
A net is any function of the form $n:(P,\le)\to X$ where $(P,\le)$ is a (preordered) directed set.
A net $m:(P',\le)\to X$ is a subnet of the net $n:(P,\le)\...
user31967
2
votes
0
answers
439
views
Quotients of simplicial complexes which are simplicial complexes
In the category of topological spaces, I would like to know that quotients of simplicial complexes (or $\Delta$-complexes) by equivalence relations which are "unramified" in a suitable sense still ...
- 31
-4
votes
1
answer
8k
views
How to transform a plane into a sphere? [SOLVED] [closed]
Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ...
- 555
1
vote
1
answer
199
views
A countable tight topological group where every countable subset is metrizable
I am looking for an example of a topological group with countable tightness with the property then it is not metrizable, but every countable subset is metrizable but I cannot construct an example.
...
- 987
7
votes
2
answers
594
views
Computational cost of converting between 3-manifold presentations
Given a 3-manifold presented as a triangulation, a Heegaard splitting, or a Dehn surgery, what is the computational cost of converting to the other two presentations? I would like Heegaard splittings ...
- 135
2
votes
0
answers
106
views
Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
- 6,700
1
vote
1
answer
386
views
Sober topological subspace
Assume $X$ to be a Notherian topological space such that any irreducible closed
subset has a unique generic point. Consider $Y\subseteq X$ as a topological space with the induced topology from $X$. Is ...
- 21
1
vote
1
answer
997
views
An open problem on general topology
There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.
Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a $...
- 654
2
votes
0
answers
178
views
Monoid prime ideals and prime congruences
I was wondering what the connection is between the notion of "prime congruence" on a monoid, and the notion of "prime ideal" in a monoid. Starting from a prime ideal $P$ in a monoid $M$, one can ...
- 4,565
8
votes
0
answers
231
views
Topology of family of complex varieties
It seems to be an oft-cited fact (which comes up, for instance, in describing vanishing/nearby cycles) that:
For a proper flat map $f \colon X \rightarrow \Delta$, where
$X$ is a complex algebraic ...
- 2,809
3
votes
2
answers
716
views
Separation axioms
Reading about separation axioms, I wonder:
Is there a separation axiom weaker than $T_2$ but stronger than $T_1$? $T_{1.5}$?
I suppose there are some separation axioms stronger that $T_6$, how many ...
- 531
2
votes
2
answers
458
views
A Fixed point Theorem that does not need the convexity of set valued map?
I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with ...
- 383
9
votes
0
answers
624
views
Two questions about universally measurable sets
I have two questions about universally measurable sets:
(1) Is there a universally measurable set of reals which does not have the Baire property?
(2) Is there a universally measurable set of reals ...
- 9,641
2
votes
3
answers
975
views
Finitely generated monoids are finitely presented?
I saw in the answer of this post that any finitely generated monoids are finitely presented in the sense that there is a coequalizer diagram $P_1\rightrightarrows P_0\rightarrow M$ with $P_1$ and $P_0$...
- 5,052
1
vote
0
answers
130
views
Is $\mathcal{P}(\omega)/(fin)$ with the interval topology path-connected?
Given a poset $(P,\leq)$ the interval topology on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\...
- 48.9k
11
votes
0
answers
404
views
Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality
A space is linearly Lindelöf iff every open cover $C$ has a subcover $S$ with $\operatorname{cf} (|S|)= \aleph_{0}$.
Question. Is there a linearly Lindelöf space $X$ with
$\operatorname{cf} (L(X))...
- 627
3
votes
0
answers
103
views
Separation-free topological completeness notion
Cannot really claim that I have immediate urgent motivation to study this question but it appeared to me long ago, I recalled it now by some reason and decided to ask it here.
There is a strong ...
- 18.4k
1
vote
2
answers
631
views
A conjecture on closed discrete subset
I am struggling with this old problem, which is also posted here:
Let $X$ satisfy countable chain condition(abbreviated as CCC) and $X$ has a regular $G_\delta$-diagonal. Then the cardinality of $X$...
- 654
3
votes
1
answer
966
views
When is the support of a Radon measure separable?
Let $X$ be a topological space, equipped with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon probability measure on $(X, \mathcal B(X))$. Recall that the support of the ...
- 8,512
2
votes
1
answer
117
views
The pseudo-metric and linear orders
Is there a necessary and sufficient condition for a linearly ordered topological space to be pseudo-metrizable? (by a pseudo-metric, I mean a map $X\times X\rightarrow\mathbb{R}$ in which all the ...
- 297
2
votes
1
answer
208
views
Does the countable $\sigma$-product of a separable Hilbert space have a first countable topology?
Let $\mathcal{l}^2$ be "the" separable real infinite dimensional hilbert space, e.g. the space of square-summable sequences of real numbers.
Let $\Box^{\mathbb{N}}\mathcal{l}^2$ be the countable ...
- 21
8
votes
2
answers
499
views
Refining open covers in locally path connected spaces
Suppose $X$ is a locally path connected topological space and $\mathcal{U}$ is an open cover of $X$ (consisting of path connected sets if we want).
One often wants the intersection $A\cap B$ of ...
- 7,193
7
votes
3
answers
1k
views
The continuous as the limit of the discrete
Reading this documment: www.math.ucla.edu/~tao/preprints/compactness.pdf, I got interested in the following thing: "One can also use compactifications to view the continuous as the limit of the ...
- 153
7
votes
1
answer
772
views
Maximal ideals of the rings of Baire-One Functions
A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all $x\...
- 1,788
2
votes
1
answer
131
views
Shrinkable decompositions with uncountably many non-degenerate elements?
Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/_{\mathcal D}$ ...
- 312
6
votes
1
answer
199
views
Constructing a odd homeomorphism between $A$ and $S^n$
I have asked this question here seven months ago and until now I got no answer.
Let $A\subset\mathbb{R}^N\setminus\{0\}$ be a closed symmetric set ($x\in A$ then $-x\in A$). Suppose that $A$ is ...
- 83