All Questions
5,184 questions
4
votes
0
answers
154
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Is there a notion of "locally flat" for CW complexes?
A submanifold $X^n\subset Y^m$ is locally flat if each point $x\in X$ has a neighborhood $U\subset Y$ so that $(U,U\cap X)\simeq (\Bbb R^m, \Bbb R^n)$ with the standard embedding $\Bbb R^n\...
- 13.6k
3
votes
0
answers
90
views
Versions of the Fréchet–Urysohn property
Recall that a topological space is called Fréchet–Urysohn if every convergent net contains (as a set) a sequence, which is convergent to the same limit. I want to refine this property as follows.
Let $...
- 5,529
24
votes
2
answers
2k
views
Why are extremally disconnected spaces so hard to give examples of?
Recall that an extremally disconnected space is a Hausdorff topological space in which the closure of any open set is still open.
On the surface, this doesn't seem like a very remarkable condition ...
- 32.5k
3
votes
1
answer
161
views
How to properly define a slice knot (or a locally flat disk)?
A knot $K\subset\Bbb S^3=\partial \Bbb D^4$ is said to be (topolopgically) slice if there is a locally flat disk $D\subset\Bbb D^4$ with $\partial D=D\cap \Bbb S^3=K$. As far as I understand, locally ...
- 13.6k
3
votes
0
answers
129
views
Topological interpretation of the existence part of the valuative criterion for properness
Let $X$ be a complex analytic space. I am trying to understand if there is a topological counterpart to the existence part of the valuative criterion for properness. The latter reads: every (ADDED: ...
- 283
6
votes
1
answer
360
views
On connected sum of compact manifolds along a submanifold
Let $M_1$ and $M_2$ be two compact manifolds of dimension $n\ge 3$. Let us have embeddings $i_1: K \to M_1$ and $i_2: K \to M_2$ for a closed manifold $K$ of dimension at most $n-1$ such that the ...
- 506
5
votes
0
answers
185
views
Stone–Weierstrass theorem for topological fields
It was showed in "The Stone–Weierstrass Theorem for valuable fields" that the Stone–Weierstrass theorem holds for any topological field whose topology comes from an absolute value or a Krull ...
- 51
3
votes
1
answer
177
views
Compactness of set of measurable functions between compact subspaces of real numbers
Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
- 131
7
votes
0
answers
349
views
An open set which is not the union of a closed set and a countable set
The following fact is probably a known result:
Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set.
Proof:...
- 1,511
4
votes
0
answers
148
views
Isomorphism between the reduced C*-algebra of a groupoid and the crossed product of inverse semigroups
In Paterson's book "Groupoids, Inverse Semigroups and their Operator Algebras" he proves that for any r-discrete groupoid $G$ with unit space $G^0$, its full $C^* $-algebra $C^* (G)$ is ...
- 271
15
votes
1
answer
602
views
Topological spaces in which countable intersections of dense open sets have dense interior
In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.
Now consider the following strengthening of the Baire ...
- 32.5k
5
votes
0
answers
192
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
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
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 ...
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 ...
- 605
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 $T^{(2)}$...
- 31
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 $k_3$-...
- 1,392
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 ...
- 1,211
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
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$.
- 313
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
views
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{...
- 2,303
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. ...
- 1,211
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 ...
- 2,826
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 ...
- 605
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
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$,...
user531303
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
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 ...
- 7,193
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
3
votes
2
answers
142
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
...
- 1,211
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
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 ...
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 ...
user530766
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,\...
- 275
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
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 \...
- 398
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
4
votes
0
answers
107
views
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
views
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
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 $B$...
- 213
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 ...
- 327
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 ...
- 742
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
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 ...
- 23
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
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^\...
- 1,442
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 ...
- 45k