All Questions
825 questions
9
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0
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569
views
A standard name for a function satisfying the intermediate value theorem?
Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
- 42k
9
votes
0
answers
367
views
A $\mathsf{ZF}$ example of two Baire spaces whose product is not Baire?
Motivated by this question I'm looking for a pair of Baire topological spaces whose product is not Baire and whose construction does not need the axiom of choice.
The example of such spaces I'm ...
- 3,011
9
votes
2
answers
2k
views
What is the free monoidal category generated by a monoid?
In several places in a segment on cohomology (for example, here (PDF)) in John Baez's online lecture notes for a course in 2007 on quantum gravity, much is made of the fact that the simplex category $...
- 1,446
9
votes
1
answer
582
views
Is there a linearly Lindelöf non-Lindelöf $P$-space?
A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersections).
A topological space $X$ is linearly ...
9
votes
1
answer
918
views
A Besicovitch-type Covering Theorem
In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...
- 1,245
9
votes
1
answer
322
views
What is the (genuine) name for the Gutik hedgehog?
Working with non-regular topological semigroups, my collegue Oleg Gutik discovered a special space $H$ which we named Gutik's hedgehog. It is homeomorphic to the space
$$H:=\{(0,0)\}\cup\{(\tfrac1n,0):...
- 42k
9
votes
1
answer
1k
views
Is the Milnor construction contractible
Let $G$ be any topological group. Then we can form the infinite join $E_G$ of $G$, i.e. the colimit $G*G*G\cdots$.
Is $E_G$ contractible?
I mean it is clear that $E_G$ is weakly contractible, but ...
- 2,554
9
votes
1
answer
437
views
Existence of a non-null-homotopic simple closed curve
Assume that $X$ is a path-wise connected Hausdorff space, and assume that its fundamental group is non-trivial. Does it always exist a simple curve in $X$ which is non-null-homotopic?
Such curve does ...
- 2,933
9
votes
2
answers
755
views
Is limit of null-homotopic maps null-homotopic?
The question is motivated by my failed comment to this one.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds).
Let $\...
- 5,529
9
votes
1
answer
602
views
On the cardinality of perfect spaces with the countable chain condition
QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces?
Recall that a ...
- 4,413
9
votes
1
answer
226
views
Is $\beta\mathbb N$ a unique compactification with the smallest possible permutation group?
For a compactification $c\mathbb N$ of $\mathbb N$ let $\mathcal H(c\mathbb N,\mathbb N)$ be the group of homeomorphisms $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\...
- 42k
9
votes
1
answer
726
views
Uniform Embedding into Euclidean Space
Given a locally compact, separable, metric space $X$.
When does $X$ uniformly embed into some Euclidean space?
This means, when does there exist some integer $n$ and a closed subset $Y\subset\...
- 3,497
9
votes
0
answers
211
views
Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?
Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones.
We can make $\textbf{Top}$ into a simplicially enriched category as follows:
Given topological spaces $X$ and $Y$,...
- 15.9k
9
votes
1
answer
1k
views
Are there intuitively clear and not technical proofs of homotopy excision theorem?
The proof given in May's "A concise course in algebraic topology", for instance, is not very involved, but quite technical. Are there less technical, but more "ideologically profound&...
- 99
9
votes
2
answers
239
views
Hausdorff open image of a Polish space
Let $f\colon X\to Y$ a continuous open and surjective function, where $X$ is Polish.
It is known that $Y$ is Polish if: $f$ is closed or $Y$ is metric.
Suppose that we know that $Y$ is Hausdorff, ...
- 313
9
votes
3
answers
1k
views
Structure Theorem for finitely generated commutative cancellative monoids?
Is there a Structure Theorem for finitely generated commutative cancellative monoids?
Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, ...
- 63.1k
9
votes
3
answers
1k
views
Relatively countably compact subsets without countably compact closure.
I am looking for a (Hausdorff or better) space $X$ and a subset $A$ of $X$ that is relatively countably compact (every sequence from $A$ has an accumulation point in $X$) such that the closure of $A$ ...
- 5,407
9
votes
2
answers
772
views
Surreal compactness
In a comment here, Joel David Hamkins said:
...I think perhaps every set-sized open cover of a bounded interval in the surreals has a finite subcover, but there are proper class open covers with no ...
- 41.1k
9
votes
2
answers
699
views
Potential connected non-Lie subgroup
This painful question is inspired by the question
"non-Lie subgroups" . Let $f$ be a discontinuous additive map from $\mathbb{R} \to \mathbb{R}$. Is it possible that the graph of $f$, inside ...
- 156k
9
votes
1
answer
1k
views
A question concerning separate and joint continuity of bilinear maps
Suppose that $V$ is a locally convex topological vector space and $f:V^2 \to V$ is a bilinear map. Suppose that $C \subseteq V$ is compact and convex, $f$ maps $C^2$ into $C$ and
$f \restriction C^2$ ...
- 3,547
9
votes
1
answer
401
views
Meager subgroups of compact groups
Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
- 1,338
9
votes
1
answer
2k
views
The generalization of Brouwer's fixed point theorem?
Let $X$ be a contractible compact [edit: locally connected] topological space
(Hausdorff and second countable). Let $f\colon X\to X$
be a continuous map. Then (I suppose) $f$ has a fixed
point. ...
- 6,891
8
votes
2
answers
579
views
Totally disconnected subspaces
This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then.
Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it ...
- 81
8
votes
2
answers
409
views
Approximation of the identity by simple functions
Let $X$ be a topological space. Assume that there exists a sequence of simple functions $\phi_n:X\to X$ (finite range and measurable) with $\lim\phi_n(x)=x$.
Can we concluded $X$ may be written by a ...
- 4,058
8
votes
1
answer
627
views
Space filling curve whose all level sets are finite (countable)
Is there a continuous surjective function $f:[0,1] \to [0,1]^2$ such that
every level set $f^{-1}(y)$ is a finite set? If the answer is no, what about if we replace the finiteness of level sets by "...
- 356
8
votes
1
answer
708
views
Immersion of $S^1$ in $\mathbb{R}^2$ that can be extended to $\mathbb{D}$
I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow \...
- 2,696
8
votes
0
answers
838
views
Intersections of open sets and $\alpha$-favorable spaces
I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...
- 4,713
8
votes
0
answers
6k
views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
- 8,512
8
votes
1
answer
716
views
Topological fraction rings and fields
Linked to this question
and as a sequel to my answer of it.
Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$.
Let
$$
s_{frac}\ :\ R\times S\to S^{-...
- 3,738
8
votes
1
answer
523
views
Are the following subsets of a Hilbert space always homeomorphic?
Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
- 724
8
votes
0
answers
285
views
Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 4,943
8
votes
2
answers
1k
views
Trivial fiber bundle
Suppose $(E,p,B;F)$ is a fiber bundle such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial?
A non connected counter example has been provided, so I'll ask for E,B ...
- 2,751
8
votes
1
answer
483
views
VC dimension of standard topology on the reals
Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is an open set $U$ with $D=S\cap U$?
I'm asking merely out of curiosity, but I'll mention that ...
- 24.8k
8
votes
1
answer
393
views
Where can I find the following S. Shelah's paper?
I've been trying to find the following article: "S. Shelah, Remarks on cardinal invariants in topology, General topology Appl. 7(3) (1977), 251-259". I tried to go directly to the journal ...
- 674
8
votes
2
answers
398
views
When are the zero sets of two continuous functions in the Stone-Čech compactification included in one another?
Let $X$ be a topological space (feel free to add some simplifying assumptions here, like “completely regular” provided at least the case of finite-dimensional manifolds is covered). Let $f,g \in C^*(...
- 32.5k
8
votes
1
answer
278
views
What is known about these "explicitly represented" spaces?
Apologies if this is too low-level. A related question that I asked on the Math Stack Exchange got no answers after a year, so I thought it might be better to ask this one here.
The standard approach ...
- 3,612
8
votes
1
answer
238
views
Functions over monoids which factor in two different ways
This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there.
Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...
- 32.1k
8
votes
1
answer
540
views
Are representations in computable analysis the equivalent to countably-generated condensed sets?
This is the first in a pair of questions. For the other see here.
Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, ...
- 6,287
8
votes
1
answer
264
views
Does the continuous image of a disc contain an embedded disc?
Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
- 13.6k
8
votes
0
answers
251
views
+300
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 501
8
votes
2
answers
592
views
Base change for category objects in topological spaces
I was prompted by this question, but the motivation is different.
Suppose we have an internal category object in topological spaces, i.e. an object space X and a morphism space Y, together with ...
- 52.6k
8
votes
5
answers
1k
views
Examples of left reversible semigroups
I am looking for concrete examples of cancellative, left reversible semigroups. Left reversible semigroups are also called "Ore semigroups". See this wikipedia page for the definition of a left ...
- 550
8
votes
2
answers
489
views
Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
- 7,179
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
8
votes
4
answers
749
views
A question about local connectedness
Let C be a connected and completely metrizable subset of the Euclidean plane. Can C fail
to be locally connected at each of its points?
- 6,215
8
votes
2
answers
362
views
Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?
Problem. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G/H$ is contractible. Is $G/H$ homeomorphic to a Euclidean space $\mathbb R^n$ for some $...
- 42k
8
votes
4
answers
6k
views
Connectedness and the real line
It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be ...
- 3,830
8
votes
1
answer
3k
views
Homotopy equivalence from contractibility of fiber
Suppose $X$ and $Y$ are two $CW$ complexes and $f:X\rightarrow Y$ is a continuous surjection such that fiber of each point (i.e. $f^{-1}(y)$ for each $y\in Y$) is contractible. Does it implies that $...
- 1,713
8
votes
1
answer
1k
views
Beautiful examples of arc-like continua
A continuum is a nonempty compact, connected metric space.
A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that $...
8
votes
3
answers
747
views
How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]
Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?