All Questions
5,184 questions
7
votes
2
answers
435
views
Is the category of quotient of countably based topological spaces cartesian closed ?
In "Handbook of categorical algebra Vol 2" from Francis Borceux, the author gives a proof that $Top$ is not cartesian closed. It seems to me that this proof can be adapted to show that the category $\...
- 764
4
votes
0
answers
331
views
What is the pro-algebraic completion of the free semigroup on one generator?
This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view.
Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...
- 38.6k
36
votes
3
answers
2k
views
Does Euclidean space have a compact factor?
Is $\mathbb{R}^n$ homeomorphic to a product $X \times Y$ with $X$ compact and not a point?
Bing's Dogbone space is a quotient of $\mathbb{R}^3$ with fibers points and arcs, and whose product with $\...
- 10.6k
7
votes
0
answers
2k
views
Has n^2*|sin(n)| limit? [closed]
Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity.
In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is ...
- 81
5
votes
2
answers
292
views
Simultaneously minimizing intersections
This may be a standard problem in homotopy theory, but I don't know a good reference.
Let $\Sigma$ be a smooth, oriented surface, and let $X_1,X_2$ and $X_3$ be three smoothly embedded curves in $\...
- 13k
2
votes
3
answers
2k
views
A question about compact subsets of Hilbert space
Let H be a separable and infinite-dimensional Hilbert space and let B be the closed ball
of H having unit radius, whose center is at the origin h of H. Suppose one would like to
know how much of B can ...
- 6,215
1
vote
1
answer
163
views
Term for number of crossings of smooth curves
Two smooth oriented finite curves $g_1, g_2$ on e.g. the 2-dimensional torus can intersect each other transversally in two ways: either the pair $(Tg_1(x),Tg_2(x))$ of tangent vectors in the ...
2
votes
1
answer
301
views
Is there existing terminology for this technical condition on semilattices?
Given a semilattice $S$, a subset $E$, and a positive integer $n$, let $E^{[n]}$ be the set of all products of $n$-tuples in $E$. Thus $\bigcup_{n\geq 1} E^{[n]}$ is nothing but the subsemigroup of $S$...
- 25.8k
15
votes
2
answers
1k
views
Exact sequence of monoids
What is the right definition of an exact sequence of monoid homomorphisms?
I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6,
http://www.math.ucla.edu/~balmer/Pubfile/...
- 3,009
10
votes
5
answers
2k
views
Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?
In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary ...
- 1,105
3
votes
1
answer
494
views
Why the category of core-compact spaces with continuous maps is not cartesian closed ?
According to ESCARDÓ-LAWSON-SIMPSON paper 'Comparing cartesian closed categories of (core) compactly generated spaces' The following four propositions are true:
A topological space $X$ is ...
- 764
3
votes
2
answers
650
views
Continuity/measurability of a complicated extension of a family of continuous functions
Bonjour/bonsoir à tous et à toutes.
I've two questions related to something on which I'm working. I've already tried to discuss about them elsewhere, but it hasn't been fruitful so far.
Edit (4 Dic ...
- 10.5k
2
votes
0
answers
167
views
Local cartesian closedness in the category of compactly generated spaces
According the the nLab, the category of compactly generated (CG) spaces is not locally cartesian closed.
So if $A$ is a CG space and $C$ a CG space above $A$, $C$ may not be exponentiable.
What if we ...
- 3,033
6
votes
0
answers
366
views
Whitney approximation without second countable
One version of Whitney's approximation theorem states the following:
Let $N$ be a smooth, Hausdorff, second-countable (or paracompact) manifold, then given any continuous function $F:N\to \mathbb{R}...
- 39.1k
4
votes
1
answer
1k
views
A boundary-preserving map on the unit disk
We are given a (closed) ball $D^n$ and a (continuous) map $f: D^n \to D^n$, that is identity on the boundary of $D^n$.
Let $C$ be a subset of $D^n$, and let $f^{-1}(C)$ be the inverse image of $C$ ...
- 41
4
votes
1
answer
921
views
Convergence in probability only depends on topology?
Suppose $(S,d)$ is a Polish space, and $X$, $(X_n)$ are random variables such that $X_n \to X$ in probability in $(S,d)$. Now suppose $d'$ is another metric on $S$, giving the same topology. Does $...
- 2,895
1
vote
1
answer
131
views
Conditions under which a given scheme converges
I'm sorry in advance for how long this question is. Suppose I have a continuous function $f:\mathbb{R}^n \rightarrow \Delta_{n-1}$, where we think of the simplex $\Delta_{n-1}$ as the set
$\Delta_{n-...
- 645
53
votes
3
answers
8k
views
Grothendieck's manuscript on topology
Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (...
2
votes
1
answer
274
views
Does X have any diagonal properties?
Assume that $2^{\omega_1}=2^\omega=\mathfrak{c}$. Let $D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$={$\xi<\mathfrak{c}:y(\xi)=1$}, the support of $y$, and let ...
- 654
5
votes
0
answers
2k
views
Is the radical of a homogeneous ideal homogeneous?
Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this generality,...
- 1,811
12
votes
1
answer
744
views
Is the following construction of the 0-Hecke monoid (well) known?
Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If ...
- 38.6k
5
votes
0
answers
308
views
Properties of the Zariski-Riemann topology on the set of valuations
One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...
- 51
9
votes
2
answers
3k
views
Topological proof of the Compactness Theorem in propositional logic without the Axiom of Choice
There is a well-known proof of the Compactness Theorem in propositional logic which uses the compactness of the space $\{0,1\}^P$, where $P$ is the set of propositional variables in consideration. In ...
- 63.1k
7
votes
1
answer
722
views
How is called a semigroup...
Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?
- 3,110
4
votes
1
answer
1k
views
Applications of Eckmann-Hilton argument to topology
There have been a couple of posts and questions on MathOverflow about the proofs of the following two facts:
Fact 1: if $X$ is a topological space, then $\pi_k(X,x)$ is abelian for $k\ge 2$.
Fact 2: ...
- 10.9k
11
votes
2
answers
811
views
Higher dimensional Heegaard splittings?
Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing ...
- 732
4
votes
1
answer
399
views
If a topological space X has $\aleph_1$-calibre, then it must be star countable?
If a topological space X has $\aleph_1$-calibre[definition], then it must be star countable?
What if the cardinality of the topological space X is additionally < = $2^{\aleph_0}$?
- 654
72
votes
9
answers
9k
views
What is a continuous path?
I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're already getting ...
- 6,034
7
votes
1
answer
2k
views
Minimize Energy for Charge Distributions
I am considering [positive] charge distributions $\rho:M\rightarrow\mathbb{R}_+$ (nonnegative reals) with unit charge $\int_M\rho=1$ for convenience. Here $M$ is a nice-enough region, say a ...
- 17.5k
5
votes
3
answers
2k
views
Very General Topology
Suppose you take mathematical structures which have axioms based on sets and their subsets and you replace this with objects and subobjects, for example:
Let a very general topological space T be an ...
- 529
2
votes
2
answers
312
views
on $F_\sigma$-discrete space
A space is $F_\sigma$-discrete space if it is the countable union of closed discrete subspaces. Is it true that every subset of an $F_\sigma$-discrete space is of the type $G_\delta$?
- 654
1
vote
2
answers
341
views
A question about connectedness in Euclidean space [closed]
Here is a question which seems true to me but I can't rigorously show. Suppose $K$ is a compact subset of $\mathbb{R}^n$ such that $\mathbb{R}^n\setminus K$ is connected, does it follow that for any ...
- 23
9
votes
0
answers
685
views
Name for a topological space where every closed set contains a closed point
A coauthor and I have stumbled upon a useful topological property -- namely, we are interested in the property that every nonempty closed set contains a closed point. However, neither of us are ...
- 1,802
3
votes
1
answer
401
views
Action on a compact group
If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?
- 1,555
8
votes
2
answers
753
views
Patching together homeomorphisms: how badly can it fail?
Suppose we have a set $X$ with $X=U \cup V$. If we pick a permutation $f$ of $U$ and a permutation $g$ of $V$ which agree on the intersection $U \cap V$, we can coalesce them into one big endo-map $F$ ...
- 3,910
19
votes
1
answer
772
views
convexity of images of space-filling curves
Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t \...
0
votes
1
answer
224
views
Special functions on the unit disk
Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.
We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a ...
- 1,271
7
votes
6
answers
1k
views
Bijective function on a dense set
Suppose X is a complete metric space, and $f:X↦X$ a continuous surjective function. Let D be a dense set. Suppose $f:D↦D$ is injective and $f^{-1}(D)=D$.
Is $f$ injective ?
Is there a family of ...
- 307
7
votes
2
answers
766
views
Question about 0-dimensional Polish spaces
Hello everybody,
I'm stuck with proving (or disproving) the following statement.
Statement:
For every $0$-dimensional Polish space $(X,\mathcal{T}\ )$, and a countable basis of clopen sets $\mathcal{...
user11618
3
votes
2
answers
510
views
Is there a countable pseudocharacter Hausdorff space，such that...?
Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two ...
- 654
4
votes
1
answer
532
views
CCC + collectionwise normality => paracompact?
Is there a CCC and collectionwise normal space, that isn't paracompact?
As we know, CCC + monotone normality => Lindelöf.
CCC + collectionwise normality => paracompact?
CCC = countable chain ...
- 654
28
votes
2
answers
2k
views
Is $\mathbb{C}^2$ homeomorphic to $\mathbb{C}^2 - (0,0)$ with the Zariski topology?
A fellow grad student asked me this, I have been playing for a while but have not come up with anything. Note that $\mathbb{C}$ is homeomorphic to $\mathbb{C} - \{0\}$ in the Zariski topology - just ...
- 12.1k
1
vote
1
answer
400
views
$G_\delta$-diagonal
Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't
a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence
${G_n}$ of ...
- 654
7
votes
5
answers
1k
views
the example of ccc but not separable
I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.
- 654
5
votes
5
answers
2k
views
Extending continuous function $D\to I$, where $D$ is a dense subspace of a separable Tychonoff space
Suppose the countable subspace $D$ is dense in the separable Tychonoff space $X$ and $f$ is a continuous function from $D$ to the closed unit interval. What are some conditions on $X$ or $D$, which ...
- 654
0
votes
1
answer
501
views
0
votes
2
answers
370
views
zeroset-diagonal
Is it true that a topology space X with a zeroset diagonal is first countable?
what if X is additionally CCC?
- 654
0
votes
1
answer
209
views
On generalized ordered spaces
Let X be a Go space. If G is open in X, why is every convex component of G open?
( It is well known that any non-void subset G of X can be uniquely represented as a union
of its maximal convex ...
- 654
14
votes
5
answers
2k
views
Largest Hausdorff quotient
The inclusion of the full subcategory of Hausdorff topological spaces into the category of topological spaces has a left adjoint, which can be proven easily by the Adjoint Functor Theorem (see for ...
- 143
6
votes
4
answers
2k
views
locally connected versus locally compact
In the definition of a locally connected space we demand every neighbourhood of a point to satisfy certain condition whereas for a locally compact space we demand that one neighbourhood be there with ...
- 508