All Questions
5,184 questions
5
votes
2
answers
310
views
A question about homeomorphic subsets of Hilbert space
Let H be a an infinite dimensional and separable Hilbert space. Let C be a closed and
bounded subset of H that is not compact. Does there always exist a closed and unbounded
subset of H which is ...
- 6,215
2
votes
1
answer
240
views
Relative extremely disconnected space
A topological space $X$ is called relative extremely disconnected if it has a base $B$ (for open subsets) such that disjoint elements in $B$ have disjoint closure.
Does it exist an infinite ...
- 192
5
votes
1
answer
452
views
Least cardinality of a set of points in the plane
What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...
- 98
1
vote
1
answer
201
views
can an nonzero IC sheaf have zero hypercohomology?
Can someone tell me which of the following are true? Let $X$ be a reasonable space.
Suppose $F$ is a complex whose cohomology groups are constructible sheaves, at least one of which is nontrivial.
...
- 8,723
3
votes
1
answer
243
views
Embedding Semigroups in Rings
Let $S$ be a finite commutative semigroup with identity. Under what conditions (on the semigroup $S$) it is possible to find a ring $R$ such that the multiplicative structure of $R - \{0\}$ is ...
- 801
-1
votes
1
answer
542
views
Fuzzy topology : references [closed]
Hey. I'm looking for references in fuzzy topology. Does anyone know a good book ?
- 11
4
votes
1
answer
448
views
Is there a name for this topology?
Let $X$ be a set and let $f: X\longrightarrow X$ be a function on $X$. Introduce a topology on $X$ by the following basis of open sets: for any subset $S$ of $X$, let $B_S$ be the set of forward ...
- 2,830
2
votes
2
answers
396
views
Closure-complement-union: countable space, finite seed, infinite family, space unique?
Let $X =$ {0, 1, 2, ...} and $T$ = { $\emptyset$, $X$, {0}, {1}, {0,2}, {0,1,3}, {0,1,2,4}, {0,1,2,3,5}, ... } $\cup$ {{0,1,2}, {0,1,2,3}, {0,1,2,3,4}, ... }. It is easily verified that $T$ forms a ...
- 870
1
vote
1
answer
189
views
constructible set and fibre product
Let $P$ be certain property. Let $S \subset \mathbb{C}^n\times \mathbb{C}^m$ be a set of closed points such that for any point in $S$, it satisfies the property $P$. I know for any $x \in \mathbb{C}^n$...
- 3,472
5
votes
2
answers
631
views
How do you know when a reflective subcategory of Top is quotient-reflective?
A subcategory $\mathcal{C}$ of the category $Top$ of topological spaces is a reflective subcategory if the inclusion functor $i:\mathcal{C}\hookrightarrow Top$ has a left adjoint $R:Top\rightarrow \...
- 7,193
3
votes
1
answer
399
views
Baire sets of $X$ possess the required Cartesian product property
Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\mid E_{i}\text{ is a Borel set in }X_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\...
2
votes
1
answer
407
views
Endomorphisms of degree d on a sphere with infinite fibers on a dense subset
Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$
disjoint n-...
- 7,609
7
votes
0
answers
2k
views
Has n^2*|sin(n)| limit? [closed]
Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity.
In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is ...
- 81
1
vote
0
answers
331
views
Relationship between weak Lp and strong Lq topologies for q<p
Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
- 203
6
votes
1
answer
555
views
Is there a name for the class of metric spaces such that the closure of the open ball of radius $r$ around each point $x$ is the set of elements $y$ such that $d(x,y)\leq r$ ?
Let $(X,d)$ be a metric space, let $B(x,r)$ be the open ball of radius $r$ about $x$ and $N(x,r)$ be the set of elements $y\in X$ such that $d(x,y)\leq r$. It is well-known that it is not always true ...
- 6,034
2
votes
1
answer
519
views
Counterexample about Jones lemma with special weak condition.
Jones Lemma is One scale about recognizing that a topological space is not normal.
This lemma tells us, if The topological space $X$ has a dense subset $D$ and a closed discrete subset $S$, with the ...
- 1,788
3
votes
0
answers
104
views
Hausdorff spaces with lattice isomorphism between the topologies [closed]
For $i=1,2$ let $(X_i,\tau_i)$ be Hausdorff spaces such that the lattices $\tau_1, \tau_2$ are isomorphic.
Does this imply that $(X_1, \tau_1) \cong (X_2,\tau_2)$?
(This is a follow-up question to $...
- 48.9k
1
vote
2
answers
412
views
When can the one-one continuous image of a perfect set fail to be perfect?
Let $\mathfrak{M}$ and $\mathfrak{N}$ be perfect Polish spaces, $P$ a nonempty perfect subset of $\mathfrak{M}$, and $f: \mathfrak{M} \rightarrow \mathfrak{N}$ a continuous surjection that's injective ...
- 1,081
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 $\...
5
votes
1
answer
723
views
Sheaf condition and representability in the category Top
This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{...
- 7,442
4
votes
1
answer
252
views
A question on hereditary Lindelof number
Let $X$ be space. A space $X$ is called right-separated if it can be well-ordered in such a way that every initial segment is open in $X$. See the related link (left-separated).
How could we show ...
- 654
10
votes
1
answer
898
views
Category Theory / Topology Question
Let me begin by noting that I know quite little about category theory. So forgive me if the title is too vague, if the question is trivial, and if the question is written poorly.
Let $\mathcal{C}$ ...
- 29.2k
4
votes
1
answer
2k
views
How much choice do we need for regularity of product of regular spaces ?
It is usually stated that the (possibly uncountable) product
of regular topological spaces is regular.
However the only proof that I know of this fact seems to use the full axiom of choice :
See ...
- 41
3
votes
1
answer
167
views
The finest countably generated free topological group so that $x^{m_{n}}_n\rightarrow 1$?
What is the finest free topological group $H$ with generators ${x_{1},x_{2},...}$ so that $x^{m_{n}}_n\rightarrow 1$ for all sequences $m_{1},m_{2},...$?
Is $H \simeq K$, with $K$ the natural ...
- 1,968
0
votes
1
answer
339
views
Thurston-Bennequin number vs. checkerboard coloring difference
For an alternating knot K, checkerboard-color the knot (if this is a lousy ASCII crossing: %, white goes to the left/right and black to top/bottom). Assume no surplus Reidemeister 1
kinks exist (K has ...
- 4,829
0
votes
1
answer
182
views
Rational points in the Alexandroff line
Let $X$ be the subset of the long line consist of rational points with the topology inherits from the long line.
Is $X$ a metrizable space?
- 356
2
votes
2
answers
809
views
On a special case of Alexander duality
Let $S^n$ be the $n$-dimensional sphere and let $K\subseteq S^n$ be a compact, locally contractible subspace of real codimension $\geq 2$. Applying Alexander duality we find that
$$
\tilde{H}_{i}(S^n-...
- 7,609
-2
votes
1
answer
395
views
non-trivial convergent sequence [duplicate]
I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$)
can you give me a example of ...
- 147
3
votes
2
answers
571
views
What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes?
For a monoid $M$ and a subset $S$ of $M$, define the syntactic congruence $\equiv_S$ of $S$ as the least congruence on $M$ that saturates $S$, i.e. :
$$u \equiv_S v \Leftrightarrow (\forall x, y)[xuy \...
- 786
0
votes
1
answer
172
views
Can we build a continuous function from "fibers"/preimages defined over a topological base?
I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...
- 273
0
votes
1
answer
91
views
Intersection of complements of connected components
Let $(X,\tau)$ be a topological space, $S\subseteq X$ such that there is $x^*\in X\setminus S$.
Let $E$ be the connected component of $X\setminus S$ that contains $x^*$. Let ${\cal C}$ be the ...
- 48.9k
24
votes
0
answers
751
views
Are amenable groups topologizable?
I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
- 3,897
5
votes
0
answers
332
views
The Haar integral on uniform spaces
Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability.
As ...
- 5,407
5
votes
1
answer
403
views
Is every bornological space measurable?
Every topological space is measurable, since we may canonically equip a topological space with its Borel $\sigma$-algebra. A bornological space is like a topological space, except the structure ...
- 8,512
3
votes
2
answers
650
views
Continuity/measurability of a complicated extension of a family of continuous functions
Bonjour/bonsoir à tous et à toutes.
I've two questions related to something on which I'm working. I've already tried to discuss about them elsewhere, but it hasn't been fruitful so far.
Edit (4 Dic ...
- 10.5k
11
votes
0
answers
422
views
Topology of marked groups for different number of generators
A $k$-marked groups is a pair $(G,S)$ where $G$ is a group and $S$ is an ordered set of $k$ generators of $G$. Each such pair can be identified with a normal subgroup of the free group $F_k$ of rank $...
- 1,378
3
votes
4
answers
627
views
Has anyone studied the applications which map open sets to either open or closed sets?
Consider two topological spaces X,Y and a function f from X to Y.
Are the following concepts already in use? How are they called?
1) f sends open subsets of X to either open or closed subsets of Y.
...
- 2,992
1
vote
1
answer
752
views
3D surfaces of infinite genus
How might one show that the set of connected 3D surfaces with infinite genus (up to homeomorphism) is countably infinite?
We could either use proof by contradiction or come up with a way to count ...
- 11
3
votes
1
answer
3k
views
Is a proper quotient map closed ?
I am trying to produce closed quotient maps, as they allow a good way of creating saturated open sets (as in this question).
A map $f:X\rightarrow Y$is called proper, iff preimages of compact sets ...
- 11.1k
5
votes
0
answers
209
views
Compact set covered by two opens
The following lemma about locally compact (but not necessarily Hausdorff) spaces or continuous lattices appears frequently but without citation. It is easy to prove but important in proofs.
If a ...
- 8,481
5
votes
0
answers
2k
views
Is the radical of a homogeneous ideal homogeneous?
Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this generality,...
- 1,811
2
votes
1
answer
122
views
Actions of compact Lie groups on (possibly but hopefully not) non-regular spaces
Suppose $G$ is a compact Lie group acting freely on a topological space $Q$ (about whose separation conditions I make no assumptions) and the qoutient $Q/G$ is known to be completely regular Hausdorff ...
- 15.5k
1
vote
1
answer
100
views
Name for (function, set) pairs?
Right now I'm working on a topological graph theory problem. To prove a theorem I introduced some objects. Has anyone heard of something similar before? I would like to call them by the right name.
...
- 1,329
9
votes
0
answers
760
views
Characterization of Unusual Topologies of $\mathbb R$
Following some argument over a question on math.SE, I began to wonder:
We all know that in the standard topology there are no continuous bijections from $[0,1]$ to $(0,1)$ (for example by arguments ...
- 39.8k
1
vote
1
answer
194
views
Modern reference request concerning Efimov's "On dyadic spaces"
Is there any modern reference (book, textbook, monograph, etc.) that contains the following result of B. Efimov (On dyadic spaces // Dokl. Akad. Nauk SSSR 151 (1963) (Russian). English translation: ...
- 895
2
votes
1
answer
125
views
open subsets of boundary [closed]
Let $\bar{X}$ be a Hausdorff and compact topological space. Suppose that $X$ is an open and dense subset of $\bar{X}$. Let $\nu X=\bar{X}\setminus X$ and assume that $U\subseteq \nu X$ is an open ...
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
0
votes
1
answer
474
views
Hilbert space having all norms (and seminorms) continous.
Suppose I have a Hilbert space $H$ such that every seminorm on $H$ is continuous with respect to the inner-product induced norm. Is $H$ necessarily finite-dimensional? If not, is there an easy ...
- 617
3
votes
0
answers
223
views
Semidirect products of semigroups [closed]
Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$.
A function $f:S\to\...
- 131
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 ...
- 356