All Questions
5,183 questions
7
votes
2
answers
1k
views
Unusual space-filling curve
Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve.
Consider a foliation as a collection of continuous nonintersecting curves that start at $(0,0)$ and ...
- 1,089
5
votes
2
answers
740
views
Conditions on a metric space so that boundedness implies total boundedness
I'm interested in conditions on a metric space $X$ which imply that boundedness is equivalent to total boundedness (or, assuming that $X$ is complete, that compactness is equivalent to precompactness)....
- 4,874
3
votes
2
answers
1k
views
Do Smash Products and Quotients Commute?
Let $X$ be a subcomplex of a CW-complex $Y$. Is $(Y/X)^{\wedge k}$ homotopy equivalent to $Y^{\wedge k}/X^{\wedge k}$, where $\wedge k$ is the $k$-fold smash product? I know it is not true for ...
- 61
2
votes
1
answer
381
views
A question about disconnecting a Euclidean space or a Hilbert space
By a "totally disconnected" point set I mean one whose only connected subsets are singletons. Can a finite dimensional Euclidean space whose dimension is at least two,
be separated by any subset that ...
- 6,215
2
votes
3
answers
975
views
Finitely generated monoids are finitely presented?
I saw in the answer of this post that any finitely generated monoids are finitely presented in the sense that there is a coequalizer diagram $P_1\rightrightarrows P_0\rightarrow M$ with $P_1$ and $P_0$...
- 5,052
1
vote
1
answer
1k
views
A question about unbounded connected subsets of the plane.
A number of clever examples have been given of unbounded connected subsets of the
Euclidean plane containing no infinite bounded subsets that are connected. None of those that
I have seen are ...
- 6,215
13
votes
4
answers
1k
views
Can Lipschitz maps increase the Lebesgue dimension ?
Given a map $f:X\rightarrow Y$ of compact metric spaces, such that there is a $C\in \mathbb{R}$ with $d(f(x),f(x'))\le C\cdot d(x,x')$.
Does this already imply, that the Lebesgue dimension of $f(X)$ ...
- 11.1k
6
votes
6
answers
2k
views
Topologically homogeneous space?
I want to know an example of a topological space $X$ which satisfies the following.
for all points $x,y\in X$ and neighborhoods $U_x$, $U_y$, there exist neighborhoods $U'_x\subset U_x$, $U'_y\subset ...
- 481
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
33
votes
6
answers
13k
views
Is a topology determined by its convergent sequences?
Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the ...
- 543
5
votes
1
answer
2k
views
How come nowhere dense subsets implies discrete?
Hi, I am reading an article and have encountered a remark in a proof which is not clear to me.
Maybe someone can help?
The proposition is:
Let X be a topological space without isolated points having ...
- 89
6
votes
3
answers
2k
views
Sequential topological vector spaces
Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree (...
- 9,650
1
vote
2
answers
504
views
Do all graphs of C1 functions have Hausdorff dimension 1?
Suppose f is a real-valued function of one variable, and suppose f is of differentiability class C1. My question is, if $\Gamma$ is the graph of f, then must $\dim_H(\Gamma)=1$? If anyone knows of a ...
9
votes
1
answer
1k
views
When completion of locally compact length space is locally compact?
As far as I know the answer to the question:
"Is it true that a completion of a locally compact length space is locally compact?" - Negative.
Does anybody know some metric and/or topological ...
- 141
2
votes
1
answer
307
views
Name for a kind of topological property?
What should I call a property (P) of (open) subspaces of a space $X$ such that:
If $U$ satisfies (P), then so does every open subset $V\subset U$
If {$U_i$} is a pairwise disjoint collection of ...
- 12.5k
2
votes
0
answers
299
views
Uniqueness of dimension for topological vector spaces
Let $V$ be a complete Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ satisfying
.
Linearly Independent: For all functions $f$ in $\mathbb{...
user5810
0
votes
1
answer
322
views
Topological Properties of Non-Smooth Functions [closed]
I am interested to know what topological properties are possessed by non-smooth functions. I suspect this is well known, but I can't find what I am looking for from google.
Is there a topological "...
- 137
12
votes
3
answers
2k
views
Minimal Hausdorff
A Hausdorff space $(X,\tau)$ is said to be minimal Hausdorff if for each topology $\tau' \subseteq \tau$ with $\tau' \neq \tau$ the space $(X,\tau')$ is not Hausdorff.
Every compact Hausdorff space ...
- 48.9k
9
votes
2
answers
534
views
A question about the dispersion points of connected metric spaces
Let $C$ be an infinite, separable and connected metric space. If $C$ becomes totally disconnected when one of its points $p\in C$ is removed, does every closed ball of $C$ with
positive radius and ...
- 6,215
27
votes
3
answers
2k
views
When does a Galois connection induce a topology?
Let $(X,\leq)$ and $(Y,\leq)$ by partially ordered sets. Recall that a(n antitone) Galois connection between $X$ and $Y$ is a pair of order-reversing maps
$\Phi: X \rightarrow Y, \ \Psi: Y \...
- 65.4k
4
votes
0
answers
354
views
Terminology for topological base closed under intersection?
Is there an established or well justified terminology for a topological base that is closed under finitary intersections?
As motivation, recall these conditions on a collection of subsets of a given ...
- 2,754
4
votes
3
answers
746
views
A question about totally disconnected point sets.
Let H be an infinite dimensional and separable Hilbert space. Let C be a closed and
connected subset of H containing more than one point. Can C ever be the countable union
of closed and totally ...
- 6,215
25
votes
10
answers
5k
views
Pair of curves joining opposite corners of a square must intersect---proof?
Reposting something I posted a while back to Google Groups.
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says
"... every pair of curves in the square joining different pairs of
...
3
votes
1
answer
405
views
Topological simplicity and dense subgroups
Let $G$ be a (topologically) simple Hausdorff topological group. Let $H$ be a dense subgroup of $G$. Now throw away the topology. What restrictions are known on the structure of $H$ as an abstract ...
- 4,728
3
votes
3
answers
933
views
The difference between a sequential space and a space with countable tightness
Hi, I have recently encountered these two definitions of a sequential space and a space of countable tightness.
And I seem to have difficulty understanding what is the difference between these two ...
- 89
33
votes
4
answers
7k
views
Topology of function spaces?
Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,...
- 33.3k
14
votes
2
answers
1k
views
Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$?
Question: Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$? I'd guess that the answer doesn't depend on choice of differentiable structure, but maybe it does.
Motivation:...
- 787
2
votes
3
answers
1k
views
Hausdorff dimension: subset of $\mathbb{R}^n$ vs. boundary of this subset
Let $n$ be a positive integer.
Let $S \subseteq \mathbb{R}^n$. Is the Hausdorff dimension of the boundary of $S$ always smaller than the Hausdorff dimension of $S$?
I have not found anything ...
- 577
19
votes
2
answers
1k
views
Are there space filling curves for the Hilbert cube?
There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: ...
- 11.1k
4
votes
2
answers
226
views
Modal models as reduced products?
In model theory for standard first-order logic, one constructs a single model, a reduced product, from a collection of first-order models, together with an index set and a filter on the index set.
In ...
- 327
6
votes
1
answer
765
views
Are finite colimits of topological spaces stable under pull-back?
The category of topological spaces has a forgetful functor to set which commutes with both small limits and colimits (it has both a left and a right adjoint). Moreover Set is a Grothendieck topos and ...
- 27.5k
5
votes
0
answers
336
views
Defining a topology by means of closed subsets in a topos
In the following we fix a topos. I'll speak of sets instead of objects and of subsets instead of subobjects.
Let $X$ be a set and assume $F$ is a set of subsets of $X$ that contains $\emptyset, X$, ...
- 63.1k
58
votes
8
answers
9k
views
Is there a Whitney Embedding Theorem for non-smooth manifolds?
For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? (i.e. Can ...
- 825
11
votes
3
answers
2k
views
Permute Wada Lakes keeping the coastline intact? (still open in dim >2)
Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical ...
- 4,188
16
votes
2
answers
2k
views
Compactification of a manifold
This is just a curiosity and the question is really foggy. I'm wondering if there can exist a notion of "minimal smooth compactification" (when I say minimal I think something like adding a finite ...
- 1,727
1
vote
1
answer
492
views
Сomplete homogeneous space which is not locally compact
It is well-known theorem that every locally compact, homogeneous, metric space is complete.
Does anybody know example of complete, homogeneous, metric space which is not locally compact?
- 141
3
votes
1
answer
394
views
Residual finiteness of groups versus residual finiteness of semigroups
A group $G$ is residually finite if, for any two elements $g$ and $g^\prime$ in $G$, there is a finite group $G^\prime$ and a (group) homomorphism $f: G \rightarrow G^\prime$ such that $f(g)$ doesn't ...
- 155
7
votes
4
answers
4k
views
Injective maps $\mathbb{R}^{n} \to \mathbb{R}^{m}$
Let $f: \mathbb{R}^{n} \to \mathbb{R}^{m}$ be an injection for $n>m$. Can $f$ be continuous? Why?
I got this question in mind when I was trying to find a continuous map from $\mathbb{R}^{2}$ to $\...
- 4,795
6
votes
2
answers
947
views
Compact cover of a Hausdorff compact space
In his book "Riemannian Geometry" do Carmo cites the Hopf-Rinow theorem in chapter 7. (theorem 2.8). One of the equivalences there deals with the cover of the manifold using nested sequence of compact ...
- 579
1
vote
1
answer
1k
views
Maximal ideals and ultrafilters [closed]
I am not sure about these two definitions.
For example, if we take the power set of A={1,2,3} with the partial order of inclusion. What are the maximal ideals and what are the maximal filters? For ...
- 11
3
votes
1
answer
321
views
Removing intersections of curves in surfaces
Let $C_1, \dots, C_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of $\...
- 32.1k
11
votes
7
answers
2k
views
Topological spaces that resemble the space of irrationals
(This question actually arose in some research on number theory.)
I once learned that any countable dense subspace of any Euclidean space $\mathbb R^n$ is homeomorphic to the rationals $\mathbb Q$.
...
- 2,858
3
votes
1
answer
528
views
A question about connected inner limiting sets
Let M be a finite-dimensional Euclidean space or an infinite-dimensional separable Banach space.
An inner limiting subset of M is a countable intersection of open subsets of M-these sets are
usually ...
- 6,215
5
votes
1
answer
378
views
Representations of products of groups (and monoids)
I have very little knowledge of representation theory, but the following has come up in my summer undergrad research project (relates to conformal field theory and geometric function theory).
Suppose ...
- 51
4
votes
1
answer
660
views
A question about Moore spaces.
Moore in his classic book "Foundations of Point Set Theory" defines a "continuum" to be a set of points
that is both closed and connected (but not necessarily compact). He also calls "non-degenerate" ...
- 6,215
0
votes
2
answers
503
views
A Jordan arc in the unit disk
Let $D$ be the open unit disk, and $J$ a Jordan arc (that is, a homeomorphic copy of $[0, 1]$) that lies in $D$, except $J(0)$ lies on the boundary of $D$, say $J(0)=1$. I would like to see that $D\...
- 95
3
votes
2
answers
571
views
What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes?
For a monoid $M$ and a subset $S$ of $M$, define the syntactic congruence $\equiv_S$ of $S$ as the least congruence on $M$ that saturates $S$, i.e. :
$$u \equiv_S v \Leftrightarrow (\forall x, y)[xuy \...
- 786
4
votes
1
answer
2k
views
How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$
For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...
- 415
3
votes
3
answers
470
views
Lebesgue dimension of images
Given a map of topological spaces $f:X\rightarrow Y$. Assume, that $X$ has finite Lebesgue dimension. I am wondering, what dim$(f(X))$ might be. Of course, if $f$ is a homeomorphism onto its image, ...
- 11.1k
4
votes
1
answer
405
views
Vocabulary on monoid periodicity
I'm learning about periodic languages, and I'm confused over the vocabulary used to describe the periodicity of (syntactic) monoids.
If I understand correctly, a monoid M is periodic if :
$$(\forall ...
- 786