All Questions
5,184 questions
0
votes
1
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278
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On the compactness of a certain chain topology [closed]
Let $X$ be a non-empty set and $I$ a collection of some nested subsets of $X$ indexed by a linearly ordered set $(\Lambda,\le)$ such that $I$ always contains the void set $\emptyset$ and the whole set ...
- 243
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
3
votes
2
answers
392
views
Ring isomorphism $\Phi \colon C(X) \to C(Y)$ and zero dimensionality of $X$
We denote the ring of all continuous real-valued functions on $X$ by $C(X)$. The ring of all bounded continuous real valued functions on $X$ is denoted by $C_b(X)$. One of the goals of the study of $...
- 1,788
1
vote
1
answer
479
views
Homology and homotopy of a surface
Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H_1(S, \mathbb Z).$
My question is; does this ...
user22741
8
votes
1
answer
1k
views
Topological necessary and sufficient condition for tightness
Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:
For each $\varepsilon>0$, we can find a compact subset $K$ of $X$...
- 3,993
16
votes
1
answer
1k
views
Does Urysohn's Lemma imply Dependent Choice?
It's widely known$^{1}$ that in the proof of Urysohn's Lemma (UL) one uses the Principle of Dependent Choice (DC). Inspired by the equivalence between DC and Baire's Category Theorem$^{2}$, I'd like ...
- 405
8
votes
1
answer
768
views
What information can one recover from the induced map on homology?
The following question came up while constructing delay embeddings of time series data.
Consider an unknown topological space $X$ and an unknown continuous function $f:X \to X$. We are given a ...
- 15.5k
4
votes
1
answer
314
views
A semigroup with the property that $x^n = a$ has at least one solution
Is there a standard name for a (multiplicatively-written) semigroup $(A, \cdot)$ such that, given an arbitrary $a \in A$, the equation $x^n = a$ has at least one solution $x \in A$ for each $n \in \...
- 10.5k
6
votes
4
answers
765
views
On Pseudo-finite topological spaces
We recall that a topological space $(X,\tau)$ is Pseudo-finite, if each compact subset of $X$ is finite.
One of the classical example of Pseudo-finite topological spaces can be considered as an ...
- 1,788
8
votes
2
answers
3k
views
Connected components of the boundary of an open subset
Hi!
Let f be a (continuous, $C^\infty$... whatever) function from $\mathbb{R}^n$ ($n \geq 2$) to $\mathbb{R}$. Assume that each connected component of $f^{-1} (0; \infty)$ and $f^{-1} (-\infty; 0)$ ...
- 1,263
3
votes
1
answer
418
views
Compact subsets and Hausdorffness of topology
We know that all closed subsets of a compact topological space $(X,\tau)$ are compact. If we add the Hausdorff condition on the topology $\tau$ we can see the equivalence of these conditions on a ...
- 1,788
3
votes
1
answer
440
views
Lindelöf subsets of $P$-spaces
A completely regular topological space $(X,\tau)$ is called a $P$-space, if every $G_\delta$-subset of $X$ is open. (i.e $\tau$ is closed under countable intersection). Here we recall some special ...
- 1,788
4
votes
1
answer
425
views
Ring structrures on R^n
Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $H$ be the set of non ...
- 7,249
0
votes
3
answers
424
views
Level 2 Menger Sponge
Hi fellows,
Does anyone know the number of holes of a level 2 Menger Sponge ?
user23855
2
votes
2
answers
408
views
When a set of measure zero plus itself contains interior
Is there a characterization of measure zero subsets $A$ of $\mathbb R^n$, $n>1$ such that the set $A+A$ contains interior? Here $A+A$ is the set of points $\{ x+y \mid x, y\in A \}$.
Is it true ...
- 415
11
votes
1
answer
997
views
How many model category structures are there on Top?
I recently started learning a little model category theory and in particular I found this nice exercise. I only know a little topology, but this prompted me to wonder how many model category ...
- 401
2
votes
1
answer
457
views
About subspaces of $F$-spaces
A topological space $X$ is an $F$-space, if Every finitely generated ideal in the ring of all continuous functions on $X$,denoted by $C(X)$, is principal. The text "Rings of continuous functions" ...
- 1,788
2
votes
1
answer
594
views
Smirnov's Deleted Sequence topology
Can anyone tell me the origin &/or original applications of Smirnov's Deleted Sequence topology? (This is #64 in Steen & Seebach's Counterexamples in Topology.) Thanks.
- 63
11
votes
3
answers
875
views
Surface Eversions: Generalizing from Sphere and Torus Eversions
In 1958, Smale proved that a $2$-sphere can be "turned inside out", and throughout the 60s, 70s, and 80s, explicit constructions such as Thurston Corrugations, and Minimax eversions were developed to ...
- 1,441
1
vote
1
answer
217
views
F-spaces and points whose complements are C*-embedded
Let $X$ be a compact Hausdorff space. If $X$ is extremally disconnected then the complement of every point $x$ in $X$ is C*-embedded in $X$ (i.e every continuous bounded real-valued function on $X\...
- 607
3
votes
1
answer
364
views
Existence of a non-submetrizable topological space $(X, \tau)$
We recall that a topological space $(X,\tau)$ is submetrizable, if there is a coarser metrizable topology $\tau'$ that $\tau\supseteq\tau'$.
one of the properties of these topological spaces is ...
- 1,788
3
votes
1
answer
261
views
Does the "measure-preserving property" commute with ultralimits ?
Let $(X, \mathcal{B}, \mu, T)$ be a measure-preserving system, with $T$ invertible, where the $\sigma$-algebra $\mathcal{B}$ is a Borel algebra arising from a topology which makes $T$ continuous, and ...
- 7,249
2
votes
0
answers
146
views
How do you call a map which sends convergent sequences to pre-compact ones ?
In my work I encountered a map $f$ between two metric spaces $X$ and $Y$ that was not continuous (at least I couldn't prove it was), but I was able to prove that convergent sequences $(x_n)$ in $X$ ...
- 4,123
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
4
votes
1
answer
1k
views
Abstract definition of properly discontinuous action
A discrete group $G$ acts properly discontinuously on a manifold $M$ if the set $\{g\in G\mid gK\cap K\neq \emptyset \}$ is finite for every compact $K\subset M$.
Is there a more abstract ...
- 1,211
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
9
votes
2
answers
1k
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Category of Uniform spaces
I suspect that the category of uniform spaces and uniformly continuous maps and the full subcategory of complete uniform spaces are both bicomplete and cartesian closed. Can anyone comfirm or deny, ...
- 582
0
votes
2
answers
210
views
Locally compact, 0-dimensional, pseudocompact space
Is a 0-dimensional, locally compact and pseudocompact space $X$ necessarily strongly 0-dimensional? I.e., must $\beta X$ be 0-dimensional?
It is known that a 0-dimensional locally compact space which ...
- 1,704
2
votes
2
answers
343
views
Does locally compact plus pseudocompact imply paracompact?
This one is probably simple, but I don't see it yet.
Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?
- 1,704
7
votes
2
answers
733
views
Existence of a compactification of $\mathbb{R}$ with $\aleph_0$ remainder
We know that the space $\mathbb{R}$ has compactifications with one point remainder, and two point remainder. but there is no compactification of $\mathbb{R}$ with three point remainder and the same ...
- 1,788
7
votes
2
answers
2k
views
A question about some special compactifications of $\mathbb{R}$
We Know that the topological space $Y$ is a compactification of the topological space $X$, if the space $Y$ is compact and hausdorff and $X$ is dense in $Y$. If for a positive integer $n$ we have a ...
- 1,788
36
votes
3
answers
10k
views
The deep significance of the question of the Mandelbrot set's local connectedness?
I am given to understand that the celebrated open problem (MLC) of the Mandelbrot set's local connectness has broader and deeper significance deeper than some mere curiosity of point-set topology.
...
- 17.6k
7
votes
3
answers
525
views
Is the class of inverse semigroups globally determined?
This question is a follow-up to this one I asked on math.stackexchange. I've decided to ask here because I believe this is a research-level question. I'm sorry if I'm wrong -- I'm not a researcher ...
- 1,445
27
votes
1
answer
2k
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Does this knot invariant distinguish trefoil chiralities?
Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$.
As a corollary of something else I was playing around with, I recently ...
- 5,191
5
votes
1
answer
458
views
Ideals of $C(X)$ with only finitely many number of zerosets
We denote the rings of all real valued continuous functions on compeletely regular Hausdorff space $X$ by $C(X)$.Let $I$ be a ring ideal of $C(X)$. define $$Z[I]:=\lbrace Z(f):\;f\in I\rbrace$$ where ...
- 1,788
1
vote
1
answer
1k
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Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question
Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
- 360
1
vote
1
answer
698
views
Bases of completely regular (Tychonoff) spaces
If the space $X$ is completely regular, we know that the collection
{${\rm int}\,Z(f)$:$f$ is a continuous function from $X$ to the real numbers} is an open base for open subsets of the space $X$ (i....
- 1,788
1
vote
2
answers
394
views
When LCS is isomorphic to subspace of some function space?
Updated: Following Michael's suggestion, I rephrase the question slightly.
Given a locally convex (Hausdorff) topological vector space (LCTVS), when is it isomorphic to a subspace of some function ...
- 101
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
4
votes
0
answers
210
views
properties of $\beta\omega\setminus\omega$ minus the P-points
Let $X=\beta\omega\setminus\omega$ and let $Y=X\setminus P$ where $P$ is the set of P-points in $X$. Then $P$ is dense in $X$ if we assume CH but $P$ may be empty otherwise (Shelah). In the one case $...
- 607
3
votes
1
answer
860
views
Counterexemple to Urysohn's lemma in a topos without denombrable choice ?
Hello !
The Urysohn's Lemma assert that in every topological spaces which is normal two closed subset may be separated by a real valued function. It's proof use axiom of countable choice (but not the ...
- 42.4k
2
votes
2
answers
328
views
non-P-points a Baire space
Let $X$ be a compact Hausdorff space. A P-point in $X$ is a point which does not lie in the boundary of the cozero set of a continuous real-valued function on $X$.
Question. Suppose that $X$ has no ...
- 607
3
votes
1
answer
502
views
Determining continuous functions on Banach spaces
Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
- 11.5k
10
votes
2
answers
1k
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Connective spectra versus simplicial abelian groups - very basic question
Hello,
I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature).
I guess that connective spectra have a model ...
- 5,562
4
votes
3
answers
329
views
closed subset of weakly lindelof
A topological space $X$ is weakly Lindelof if every open cover has a countable subfamily $U$ such that $\bigcup \{ V: V\in U\}$ is dense in X.
Question: Are closed subsets of weakly Lindelof spaces ...
- 607
3
votes
1
answer
419
views
Relation on the set of connected components of the $\mathbb{C^*}$-fixed points locus coming from the Bialynicki-Birula decomposition
Let $X$ be a smooth variety with an action of $\mathbb{C}^*.$ One has the so-called Bialynicki-Birula decomposition of $X$ given by stable manifolds: $$X=\bigcup_N X^+(N),$$ where $N$ varies in the ...
user17778
8
votes
2
answers
3k
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Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.
Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$
Let us define the following:
Condition 1: $X$ can be covered by ...
user17778
3
votes
1
answer
459
views
When is the Freudenthal compactification an ANR?
Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is:
What are ...
- 2,104
10
votes
3
answers
1k
views
Are k-spaces named for Kelley?
On page 58 of Mark Hovey's book Model Categories, he states the following definitions:
"A subset $U$ of a space $X$ is
compactly open if for every continuous
$f:K\rightarrow X$ where $K$ is
...
- 30.3k
7
votes
2
answers
473
views
Trivial cobordism group in dimensions 1, 3, 7 related to H-space structures on the spheres in these dimensions?
Is there a connection between the existence of H-space structures on $S^1$, $S^3$ and $S^7$ and the fact that every (closed) 1-manifold, 3-manifold and oriented 7-manifold is a boundary, or is this a ...
- 931