All Questions
5,183 questions
3
votes
0
answers
60
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What circumstances guarantee a p-adic affine conjugacy map will be a rational function?
Let $\Bbb Q_p$ be a p-adic field and let any element $x$ of $\Bbb Q_p$ be associated with a unique element of $\Bbb Z_p$ via the quotient / equivalence relation $\forall n\in\Bbb Z:p^nx\sim x$
Then in ...
- 308
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
5
votes
1
answer
215
views
Is it possible to fill a boundary component of an irreducible 3-manifold using a handlebody so that the resulting manifold is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with boundary (possibly more than one component). Let $S\subseteq\partial M$ be one of its boundary components, which is an orientable surface ...
- 28.2k
5
votes
0
answers
191
views
Do most semigroups have a zero?
It is widely believed in finite semigroup theory that asymptotically almost all finite semigroups $S$, up to isomorphism, are 3-nilpotent, i.e., they satisfy $\#\{abc\,:\,a,b,c\in S\} = 1$. My ...
- 151
5
votes
1
answer
380
views
Proving the Cork Theorem
I am reading Kirby's paper paper "Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds" and I have a question about how to actually prove the cork theorem from the results ...
- 1,040
9
votes
0
answers
180
views
How should we picture the set of monomial orders (= positive monoid orders on $\mathbb{N}^k$)?
Motivation: So apparently there's some sort of sport competition currently going on where I live, which leads to countries being given an element of $\mathbb{N}^3$ called a “medal count”, and not ...
- 32.5k
1
vote
1
answer
132
views
Is the product of torus and sphere a cover of the symmetric square of torus?
Let $T$ denote the $2$-dimensional torus and $T^{(2)}$ denote its symmetric square (which is the orbit space of the canonical $\mathbb{Z}_2$ action on the $4$-torus $T \times T$).
One can see $T^{(2)}$...
- 148k
1
vote
0
answers
36
views
When must a space generated by compacts also be generated by Hausdorff compacts?
Cross-posted from Math.SE: https://math.stackexchange.com/questions/4948421/.
I'm interested in comparing $k_1$-spaces,
spaces whose topologies are witnessed by
their compact subspaces, and $k_3$-...
- 1,362
4
votes
1
answer
1k
views
Original proof of the Borsuk-Ulam theorem
I am looking for the original proof by Borsuk of the Borsuk-Ulam theorem. I would appreciate very much if someone could outline the proof.
- 356
5
votes
3
answers
851
views
What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?
Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...
- 17k
1
vote
0
answers
66
views
Extending homeomorphisms on closure spaces
Let $C$ be an infinite $T_1$ closure space, which is not a topological space. Suppose $C$ has the exchange property: for $x,y\in C$ and $A\subseteq C$
$$
\big( x\notin\overline{A}, \hspace{4mm} x\in \...
- 2,605
4
votes
1
answer
183
views
When can a generalized connected sum be aspherical
Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$...
- 311
3
votes
1
answer
191
views
Extensions of bounded uniformly continuous functions
Let $X$ be a uniform space, $S\subseteq X$ and $f:S\to \mathbb{R}$ bounded uniformly continuous, then there exists a uniformly continuous extension of $f$ to $X$. (Katětov, 1951)
I am looking for ...
- 11.4k
3
votes
0
answers
159
views
A question regarding weak Whitney embedding theorem
The weak Whitney embedding theorem states that any continuous map $f: D^n \to \mathbb{R}^{2n+1}$ (Let us focus on $D^n$ for this question) can be approximated (in $C^0$-norm) by embeddings. A counter ...
- 31
3
votes
0
answers
119
views
The topological entropy of potential space filling curves on the unit interval
By a potential space filling curve we mean a continuous function $f:[0,1]\to [0,1]$ such that there is a continuous surgective function $g:[0,1]\to [0,1]^2$ with $f=\pi_1 \circ g$ where $\pi_1$...
- 356
2
votes
1
answer
95
views
Specific distance between sets of points
Let us have closed curve without self-intersections,
initial point $O$ and curve parameter $t$, $0 \leq t \leq t_{\max}$ so $t(O) = 0 = t_{\max}$.
There are two sets of points on the curve, which are ...
- 512
1
vote
0
answers
90
views
Well-embedded type property for bounded functions
According to @Tyrone the term well-embedded set was first used in Measures on Metacompact Spaces by W. Moran.
In the article Extensions of Zero-sets and of Real-valued Functions by R. Blair and A. ...
- 1,201
1
vote
1
answer
215
views
Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
- 468
14
votes
0
answers
326
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
- 3,751
18
votes
0
answers
323
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The analogy between dualizable categories and compact Hausdorff spaces
Efimov has in his recent preprint K-theory and localizing invariants of large categories, Appendix F, a long table of analogies between the categories $\text{Cat}^\text{dual}_\text{st}$ and $\text{...
- 12.9k
9
votes
2
answers
471
views
Proving the inequality involving Hausdorff distance and Wasserstein infinity distance
Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$
where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\...
- 4,713
9
votes
0
answers
258
views
Sheaf cohomology of non-paracompact manifolds (e.g. the long line)
I have long heard that manifolds are "affine". If we allow non-paracompact manifolds, then this seems to fail, since as explained in Dmitri Pavlov's answer, the Serre–Swan theorem fails. I ...
- 2,806
7
votes
2
answers
534
views
Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...
- 557
13
votes
1
answer
329
views
Is there a metric compactification that doesn't create new paths?
Every separable metric space $A$ has a metrizable compactification, i.e. a compact metrizable space $X$ for which $A$ embeds topologically as a dense subspace of $X$. There are many approaches to ...
- 12.9k
0
votes
1
answer
142
views
"Locally compact"-ly generated topological spaces
Let $P$ be a property of topological spaces - here I am interested in "compact" and "locally compact".
A topological space $X$ is $P$-ly-generated if, for any topological space $Y$,...
- 42.4k
3
votes
0
answers
124
views
Injective envelope of B(H)
$B(\ell^2)$ is an injective operator system by a result of Arveson. However, $B(\ell^2)$ is not an injective Banach space, since it is not linearly isomorphic to a $C(K)$ space (for instance, $C(K)$ ...
- 2,605
3
votes
1
answer
645
views
Is the Hopf link a Brunnian link?
I'm trying to fill a woeful gap in my topological knowledge and learn a little knot and link theory (I'll be recording my progress on the nLab, starting with a page on links). Not wishing to write ...
CommunityBot
- 1
3
votes
2
answers
141
views
Countable zero-sets are $C$-embedded?
I was browsing Gillman and Jerison for known relations between zero-sets, $C$-embedded sets and so on.
The spaces I'm considering are $T_{3.5}$.
There are two properties that pseudocompact spaces have
...
- 5,596
15
votes
1
answer
507
views
Is there an infinite subset of $\Bbb{R}$ not homeomorphic to any of its proper subsets?
Is there an infinite subset of $\Bbb{R}$ that is not homeomorphic to any of its proper subsets? Clearly, any finite subset of $\Bbb{R}$ is not homeomorphic to any of its proper subsets by mere ...
- 255
11
votes
2
answers
3k
views
“is topologically mixing” vs. “is topologically transitive” in the definition of chaos
This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours.
Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits"...
- 4,713
1
vote
0
answers
76
views
Shellable non-pseudomanifolds with dimension greater than 2
Shellability of simplicial balls and spheres (simplicial complexes whose geometric realizations are homeomorphic to balls and spheres) has been studied quite extensively. There are many explicit ...
5
votes
1
answer
104
views
When do two measured foliations on a surface define a Riemann surface structure?
Let $S$ be smooth surface of finite type, i.e. it has genus g and n punctures (assume $S$ to have negative Euler characteristic). We know by Hubbard-Masur theorem that given a measured foliation $(F,\...
- 680
1
vote
0
answers
48
views
Connected pre-images spanning $n$-cubes under dimension reducing maps
Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...
- 63.9k
2
votes
0
answers
104
views
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
3
votes
0
answers
152
views
Topological counterexample for M(K, Y1 × Z1) being a subbasis of the compact open topology of C(X,Y×Z)
We are trying to answer whether the following mapping is continuous and open
$$C(X, Y \times Z) \to C(X, Y ) \times C(X, Z)$$ (the topological spaces being provided with the compact-open topology). We ...
- 39
3
votes
1
answer
103
views
Topology on set of "real lower bounds"
Specific question: Is there a name for the "topology of real lower bounds"? This is the order topology for the ordering $\supseteq$ on the set
$$
\mathbb{LB} = \bigl\{ [t, \infty) \mid t \...
- 18.6k
3
votes
3
answers
583
views
Topology induced by context-free language
Is there any way to reasonably define a topology on a context-free-language language? In other words, given a context-free grammar (or perhaps a grammar from an interesting subclass of context-free ...
- 260
2
votes
1
answer
174
views
A topological space has the homotopy-type of a CW-complex, then is it locally contractible?
Let $X$ be a topological space which has the homotopy-type of a CW-complex. As well-known, a CW-complex is locally contractible.
Question: Is $X$ locally contractible? If not, can some one give a ...
- 5,596
5
votes
1
answer
247
views
Does a "good" homotopy equivalence between pairs imply homotopy equivalence between quotient spaces?
If $(X,A)$ and $(Y,B)$ are (good) pairs of topological spaces, and $f:X\rightarrow Y$ is a homotopy equivalence such that the restriction $f\restriction_A$ is a homotopy equivalence between $A$ and $B$...
- 11.1k
4
votes
0
answers
107
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Reference request for a theorem of Jaworowski
Jan Jaworowski, in 2000, proved the following theorem (I came to know about it from here)
Jaworowski (2000) : Let $Y$ be a finite simplicial complex of dimension $k$ and let $n\ge 2k$. If $f:S^n\to Y$...
- 141
10
votes
0
answers
159
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Closed sets versus closed sublocales in general topology in constructive math
This question is set in constructive mathematics (without Choice), such as in the internal logic of a topos with natural numbers object, or in IZF.
Short version of the question: if $X$ is a sober ...
- 32.5k
2
votes
1
answer
57
views
Are simplicial commutative inverse semigroups fibrant?
Let $X$ be a simplicial object in the category of commutative inverse semigroups (or monoids, if needed). Is the underlying simplicial set of $X$ always a Kan complex? If so, are there some nice ...
- 37.8k
2
votes
1
answer
123
views
Signed measures on algebras (fields) and their boundedness properties
I asked this question here on math.StackEchange, but it might be too technical so I re-post it here.
Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable ...
- 2,830
3
votes
0
answers
81
views
Mixing flow has aperiodic orbit?
Let $X$ be a compact connected metric space with more than one point.
Suppose that $H:X\times [0,\infty)\to X$ is continuous such that $h_0=H\restriction X\times \{0\}$ is the identity on $X$, and $h_{...
- 3,317
0
votes
0
answers
64
views
Can an upper hemicontinuous correspondence be discountinuous everywhere?
Let $\phi: X \rightrightarrows Y$ be an upper hemicontinuous correspondence. If $K \subset X$ is a compact and convex set, $K$ contains an open set $U$, and $\phi(x)$ is nonempty, compact, and convex ...
- 101
10
votes
6
answers
879
views
Countable chain condition in topology
A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. ...
- 467
5
votes
2
answers
247
views
Definability properties of box-open subsets of Polish space
Let $X$ be a perfect Polish space $X$, so that $X^\omega$ is also a Polish space under the product topology. Call a subset $\mathcal{X} \subseteq X^\omega$ box-open if it is an open subset of $X^\...
- 14k
13
votes
1
answer
355
views
Canceling $\mathbb{R}$-factor
Suppose there are compact sets $K_1,K_2\subset\mathbb{R}^n$ such that $K_1\times \mathbb{R}\cong K_2\times \mathbb{R}$,
but $K_1\ncong K_2$.
What is the minimum of $n$?
Comments
The spherical ...
- 11.1k
1
vote
0
answers
228
views
Is the topological dimension of spacetime fixed for causally isomorphic spacetimes?
Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are not homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
The Lorentzian metrics $g_1$ and $...
CommunityBot
- 1
4
votes
2
answers
292
views
$\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ or a correction to this quotient group relation
We know that there is a fiber sequence:
$$
\dotsb \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to \dotsb.
$$
Is this fiber sequence induced from a short exact ...
- 12.9k