All Questions
1,340 questions with no upvoted or accepted answers
3
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98
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How Universal is the Topological $\mathbb K$-algebra $C(\Omega, \mathbb K)$?
For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. ...
- 1,955
3
votes
0
answers
501
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Some counter examples in group theory
In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in \{1,2,\ldots,...
3
votes
0
answers
113
views
Adjunctions of uniformly locally connected spaces
A space $X$ is uniformly locally connected (ULC) if there exists an open neighbourhood $U$ of the diagonal $\vartriangle_X$ in $X \times X$ and a homotopy $H: U \times I \to X$ between $\pi_1|U$ and $\...
- 81
3
votes
0
answers
115
views
Cardinality based results in Topological Vector Spaces?
Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen:
TVS's $V$ of larger ...
- 437
3
votes
0
answers
201
views
Which compact topological spaces are homeomorphic to their ultrapower?
It is well known that for any compact metric space $(X, d)$, and any ultrafilter $\mu$ there is a map $i_\mu:\prod_\mu (X, d) \to (X_d)$ in the category of metric spaces and Lipschitz maps where $i_\...
- 2,221
3
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answers
314
views
Certain conditions on cancellative semigroups
This is extracted from this question following Benjamin Steinberg's suggestion.
For a semigroup $S,$ let $P(S)$ denote the power semigroup of $S,$ which is made up of all non-empty subsets of $S$ ...
- 1,445
3
votes
0
answers
103
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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
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402
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Generalization of Jordan Curve Theorem
Jordan Curve Theorem says that any plane continuum homeomorphic to $\mathbb{S}^1$ separates the plane into exactly two components.
Now
"Let $\alpha$ and $\beta$ be two homeomorphic plane continua. ...
- 615
3
votes
0
answers
126
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dual composition of binary relations
I'm not sure if this is of any interest at all, but I spent some time looking at it a couple of years ago so I'd like to ask for input on this.
Given two binary relations $\rho,\,\sigma$ on a set $X,$...
- 1,445
3
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0
answers
115
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Characterization of global sections (which are not products) of a sheaf which is locally a product
In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
- 397
3
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0
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316
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Order dimension vs topological dimension of a poset
Let $(P,\leq)$ be a partially ordered set (poset). We define the ordering dimension $\textrm{dim}_\textrm{ord}(P)$ of $(P,\leq)$ to be the smallest cardinal $\kappa$ such that there exist a set of ...
- 48.9k
3
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122
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A topological space extracting from a group action
Let $G$ be a compact abelian topological group with invariant measure $\mu$ which acts on a compact Hausdorff space $X$. A $G$-odd function is a continuous function $f:X\to \mathbb{C}$...
- 366
3
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0
answers
61
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Local cross sections in infinite dimensional groups
Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples ...
3
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141
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Non-finitely based varieties and pseudovarieties
The variety of semigroups defined by $B=\Big\{(x^py^p)^2=(y^px^p)^2:p \text{ is prime}\Big\}$ is non-finitely based (Isbell, 1970). Is the pseudovariety defined by $B$ also non-finitely based?
More ...
- 563
3
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303
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Pseudomodules, "general coherence theorem"
A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...
- 2,312
3
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334
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Homeomorphism of compact Hausdorff spaces
(Note: I asked this question at MSE over a day ago and received no answer, so I'm now reposting it here. Link: https://math.stackexchange.com/questions/853500/homeomorphism-of-compact-hausdorff-spaces)...
- 191
3
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answers
171
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Berkovich Analytification of the transseries
I am looking for references to articles about the following subjects:
Connections from the field of (real) transseries to the field of surreal numbers (mentioned very briefly in the introduction of ...
- 161
3
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answers
373
views
Closed Graph Theorem and Spaces Of Continuous Functions
Let $X$ be a (Tychonoff) topological space. Consider $C\left(X\right)$ being a topological vector space of all continuous scalar-valued functions with the compact-open topology.
Assume that $Y$ is a ...
- 5,529
3
votes
0
answers
174
views
Number of k-generated semigroups
Given some $k>1$, I am interested in the number of $k$-generated semigroups of order $n$ (either up to isomorphism or all associative binary operations on an n-element set). At first I thought $3$-...
- 53
3
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answers
551
views
Is the limit set of a group action always closed?
Let $G$ be a discrete group acting on a compact metric space $X$. A point $x\in X$ is called a limit point, if there is a base point $x_0\in X$ and an injective sequence $(x_k)_{k\in\mathbb{N}}$ in ...
- 2,581
3
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answers
305
views
Paracompactness and inner product on vector bundles
Let $X$ be a Hausdorff topological space. It is well-known that every vector bundle on $X$ possesses a continuous inner product, provided that $X$ is paracompact.
Is the converse true? -- Namely ...
- 366
3
votes
0
answers
107
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Hindman's theorem variant for noncommutative semigroups
The well set proof of Hindman's finite sums theorem applied to noncommutative semigroups yields a sequence of elements such that finite products ordered coherently with this sequence are in one set. ...
- 31
3
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431
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Bohr topos and quantization
Bohrification is a natural way to construct a quantum "phase space" (with some nice insights on foundational problems like non-contextuality through Kochen-Specker etc). I was wondering, since we get ...
- 585
3
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0
answers
176
views
Extending a Hilbert space isometrically
Let $H$ be a Hilbert space, and let $X$ be a topological vector space.
Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$?
...
- 8,512
3
votes
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answers
166
views
A question of terminology - Unitizations of semigroups
There are at least two standard ways of unitizing a (small) semigroup $\mathbb A$:
(i) We add an identity regardless that $\mathbb A$ is already unital.
(ii) We add an identity only if none is ...
- 10.5k
3
votes
0
answers
867
views
The inductive and projective limits of compact Hausdorff topological groups
Are there conditions known under which the inductive or projective limit of a family of compact Hausdorff topological groups is compact? (For instance, such a result is valid for the projective limit ...
- 5,407
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 $X$:
$$
V\...
- 7,609
3
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0
answers
277
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For METRIZABLE spaces, do the Banach classes and Baire classes coincide?
In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...
- 456
3
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answers
251
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What is the origin of the metrization problem for compact convex sets?
The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...
- 3,547
3
votes
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answers
168
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Mapping into Hurewicz cofibrations.
In Strøm's paper "The Homotopy Category is a Homotopy Category" he proves
(Lemma 4) that if $Y$ is compact and if $i:A\to X$ is a cofibration, then the induced map
$$
i_*: A^Y \to X^Y
$$
is also a ...
- 12.5k
3
votes
0
answers
294
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Monomorphisms in geometry
What is known about monomorphisms in the following categories:
Schemes
Complex manifolds
$C^\infty$--manifolds
and any other kinds of geometric objects that you might think of.
How do we choose a ...
- 123
3
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0
answers
126
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More on continuous images of dense orders
In this question I asked if there was an analogue of connectedness which applied to dense orders which were not required to be complete. Between them, Noah and Joel showed that every (infinite) ...
- 3,612
3
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0
answers
267
views
Maps of loop spaces with infinity-bounded differential.
I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally:
In the following piece-wise smooth means smooth on ...
- 2,590
3
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answers
721
views
What is the horn torus homeomorphic to?
Is the horn torus homeomorphic to some other well known object? In particular, the standard torus can be described by a square with collapsed edges. What about the horn torus?
- 1,638
3
votes
1
answer
440
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In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?
In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
Note that ...
- 4,862
2
votes
0
answers
83
views
Is there a natural topology for subsets of a fixed topological space?
This question is an extension/clarification of the question: Is there a natural topology for sets of topological spaces?
The Hausdorff distance assigns a distance to any two subspaces $X, Y$ of a ...
- 719
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answers
159
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About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev
According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result:
Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a
family of ...
- 649
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
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0
answers
29
views
When are canonical maps of a filtered colimit open/closed, given that the transition maps are open/closed?
Let $X_i$ be a filtered diagram of topological spaces. I am interested in when the canonical maps $f_i:X_i\rightarrow \text{colim } X_i$ are open/closed. It is pretty easy to show that if the ...
- 41
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146
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Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R}$
About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the ...
- 366
2
votes
0
answers
104
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When do filtered colimits commute with finite products in Top
It is well known that filtered colimits commute with finite products (more generally any finite limit). This is not the case in general in Top due to Top not being cartesian closed. My question is is ...
- 41
2
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0
answers
71
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Topological measure theory on spaces that are not completely regular
In the usual discourse regarding approaches to measure theory, it is often pointed out that the restriction of topological measure theory to locally compact Hausdorff spaces is insufficient. However, ...
- 3,816
2
votes
0
answers
81
views
Extension of a tangent vector field
Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
- 434
2
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0
answers
156
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Testing for weak homotopy equivalences with compact Hausdorff spaces
Let $f \colon X \to Y$ be a weak homotopy equivalence between topological spaces. If I am not mistaken, then one can rephrase this by stating that the induced map $[K,X] \to [K,Y]$ between homotopy ...
- 2,998
2
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0
answers
55
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Fundamental group of cyclic branched cover of affine plane
Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
- 615
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0
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88
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Union of two open, open-unicoherent sets whose intersection is connected
I stumbled upon the following proposition, and haven't found an error in my proof yet.
By "open-unicoherence" I mean unicoherence with closed sets replaced with open sets in the definition.
...
2
votes
0
answers
80
views
An alternative definition for finitely generated (and principal) ideals in a semigroup
Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power ...
- 10.5k
2
votes
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answers
35
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Continuity of Kernel Mean Embeddings
Given some kernel $k: X \times X \to \mathbb{R}$ with RKHS $H_k$ we say that $k$ is characteristic on the space of signed Radon measures over $X$, denoted by $\mathcal{M}(X)$, if the kernel mean ...
- 161
2
votes
0
answers
91
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A recursive description of the smallest divisor-closed subsemigroup containing a set
Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).
...
- 10.5k
2
votes
0
answers
164
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Triviality of map $(\Sigma \theta)^*$
We know that there is a cofibration sequence
$$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
