All Questions
5,184 questions
2
votes
1
answer
630
views
Knowledge base about topology [closed]
We are studying topology. There are a lot of definitions and theorems. I wonder if there somewhere knowledge base about topology and reasoning system exists. So I expect some tool that systematizes ...
- 23
36
votes
4
answers
5k
views
Compact open topology on $\mathrm{Homeo}(X)$
Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)...
- 2,751
8
votes
2
answers
1k
views
Trivial fiber bundle
Suppose $(E,p,B;F)$ is a fiber bundle such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial?
A non connected counter example has been provided, so I'll ask for E,B ...
- 2,751
2
votes
3
answers
538
views
General theory for p-normed spaces
Hello,
in functional analysis and operator theory, you encounter several (at first glance, at least) similar constructions of normed spaces that can be indexed with some $ p \in [1,\infty]$, and ...
- 5,327
2
votes
1
answer
220
views
homeomorphisms on k-spaces
Let X be a Hausdorff k-space (Hausdorff compactly generated space) and h a bijection on X such that for any subset E of X we have
...
- 23
3
votes
1
answer
419
views
Question on coverings and and their classifying spaces [closed]
Hello. I have a question on covering spaces. Please let each space be paracompact, path-connected, locally path-connected and semi-locally simply connected.
Let $E\to B$ denote a normal covering ...
- 31
10
votes
1
answer
2k
views
Is a space with no covering spaces simply connected?
Suppose $X$ is a path connected space such that every connected covering space of $X$ is trivial (1-fold.) Must $X$ be simply connected?
Intuitively, the answer seems to be no (imagine taking a disk,...
- 713
3
votes
0
answers
251
views
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
13
votes
1
answer
736
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,547
4
votes
3
answers
213
views
Cancellation of contractibility of fibres
Suppose given maps $f:X \to Y$ and $g:Y \to Z$ such that $f$ and $g \circ f$ both have contractible fibres. Then does $g$ have contractible fibres?
And, the same question, but with the maps assumed ...
- 8,723
5
votes
1
answer
294
views
Isotropic subspaces in cohomology
Hello,
Here is a problem I encountered in the study of Kähler manifolds but there is a natural generalisation of this for topological spaces.
If $X$ is a topological space, denote by $g_\mathbb{R}$ ...
- 339
14
votes
3
answers
1k
views
What spaces can be obtained from $\mathbb{R}^{n}$ by taking quotient spaces and subspaces?
Is there a good characterization of the smallest collection of topological spaces which contains $\mathbb{R}^{n}$ for each $n$, and is closed under taking subspaces and quotient spaces?
A bit of ...
- 726
9
votes
2
answers
928
views
Is there a long exact sequence associated to a ramified covering?
A covering map $p:X\to Y$ between topological spaces can be viewed as a fiber bundle $\Sigma\to X\to Y$ with a discrete group $\Sigma=Gal(X/Y)$ as fiber. Such a fiber bundle leads to a long exact ...
- 681
5
votes
1
answer
498
views
Percolation in Cayley graphs of semigroups.
Percolation in Cayley graphs of groups are studied by many researchers. There are also the concept Cayley graphs for semigroups. Are there any research about percolation in Cayley graphs for ...
- 6,201
16
votes
12
answers
5k
views
Examples of $G_\delta$ sets
Recall that a subset $A$ of a metric space $X$ is a $G_\delta$ subset if it can be written as a countable intersection of open sets. This notion is related to the Baire category theorem. Here are ...
- 18.7k
0
votes
1
answer
271
views
Numbers associated with boundaries of manifolds
I don't know what name if any is attached to the numbers I'm about to describe.
For a line segment, [a,b]
the number is 1 if for any k in (a,b)
and 2 if k=a or k=b.
For a square, [a,b] ...
- 379
0
votes
1
answer
147
views
Small set of acts over a countable monoid?
Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?
- 11
6
votes
2
answers
5k
views
On the cohomology of a finite covering map
So let $X$ be a "nice" topological space and assume that $G$ is a finite group which acts freely on $X$.
Q: Is there a simple relationship between the cohomology groups
$H^i(G,\mathbf{Z}), H^i(X,\...
- 7,609
7
votes
4
answers
2k
views
Must a linearly ordered, separable space be metrizable?
Let $X$ be a linearly ordered topological space with a countable dense subset. Does it necessarily follow that $X$ is metrizable?
EDIT: Apollo's comment int he answers implies the answer is negative. ...
- 656
0
votes
1
answer
270
views
Random question: Is there a set-theoretic description of projective space? [closed]
I met projective space via a recent class on perspective drawing, believe it or not, but I didn't know that this was the "space" we were using. I came across a more detailed description trawling the ...
- 91
10
votes
4
answers
2k
views
When do isometric actions exist?
Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
- 545
1
vote
1
answer
154
views
undecidability in the dynamics of functions $f: \Sigma^* \rightarrow \Sigma^*$
Does anyone know any undecidable problems in the dynamics of functions (not necessarily monoid homomorphisms or anything) from \Sigma^* to \Sigma^* where \Sigma is a finite set? In particular, I'm ...
- 549
16
votes
5
answers
9k
views
A G-delta-sigma that is not F-sigma?
A subset of $\mathbb{R}^n$ is
$G_\delta$ if it is the intersection
of countably many open sets
$F_\sigma$ if it is the union of countably many closed sets
$G_{\delta\sigma}$ if it is the union
of ...
- 1,981
18
votes
2
answers
1k
views
Example of a compact homogeneous metric space which is not a manifold
A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd ...
- 11.4k
3
votes
2
answers
994
views
measurability of integrated functions
DISCLAIMER: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a homework, as ...
user11618
7
votes
3
answers
3k
views
Is a connected separable locally euclidean Hausdorff topological space second countable?
This question arose from considering for a connected smooth Hausdorff manifold the (possible) equivalence of the following properties:
(1) paracompact,
(2) metrizable,
(3) second countable,
(4) ...
- 3,584
6
votes
3
answers
771
views
Null-homotopy of diagonal map
For a sphere $S^n$, the diagonal map $\Delta:S^n\to S^n\wedge S^n$ sending $x\mapsto x\wedge x$ is null-homotopic. This is the homotopy group $\pi_n(S^n\wedge S^n)=\pi_n(S^{2n})$ is trivial.
I'm ...
- 681
0
votes
1
answer
474
views
Hilbert space having all norms (and seminorms) continous.
Suppose I have a Hilbert space $H$ such that every seminorm on $H$ is continuous with respect to the inner-product induced norm. Is $H$ necessarily finite-dimensional? If not, is there an easy ...
- 617
8
votes
3
answers
2k
views
Spaces with a quasi triangle inequality
How do you call a space with a function which is symmetric, non negative, positive definite and which satisfies a quasi-triangle inequality:
$d(x,z) \leq C( d(x,y)+d(y,z) )$
for all $x,y,z$ and some ...
- 217
17
votes
12
answers
4k
views
Why semigroups could be important?
There is known a lot about the use of groups -- they just really appear a lot, and appear naturally. Is there any known nice use of semigroups in Maths to sort of prove they are indeed important in ...
- 1,437
68
votes
3
answers
21k
views
Properly Discontinuous Action
When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
- 1,236
24
votes
2
answers
4k
views
complement of a totally disconnected closed set in the plane
While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
1
vote
1
answer
362
views
Winding number bijection on graphs
Let $G=(V,E)$ be an isoradial graph. In other words the graph can be imbedded into the plane such that each face (plaquette) can be circumscribed a circle of radius 1 with the circle's center ...
- 4,952
2
votes
1
answer
321
views
CG Hausdorff space
Let X be in CGHaus and Y locally compact hausdorff. The usual product space XxY is CGHaus, so we dont need to apply that special functor to it (the one that takes a space to the space with same points ...
- 21
7
votes
6
answers
2k
views
Elegant representations of graphs in R^3
If I have a graph of a reasonable size (e.g. ~100 nodes, ~40 edges coming out of each node) and I want to represent it in R^3 (i.e. map each node to a point in R^3 and draw a straight line between any ...
- 283
6
votes
2
answers
462
views
need references regarding the elementary theory of free semigroup and free abelian groups
Recently, I read that two free abelian groups $S$ and $T$ have the same elementary theory if and only if rank$S$=rank$T$. Does anyone have a reference with a proof of this? Also, what is known about ...
- 549
4
votes
1
answer
2k
views
How much choice do we need for regularity of product of regular spaces ?
It is usually stated that the (possibly uncountable) product
of regular topological spaces is regular.
However the only proof that I know of this fact seems to use the full axiom of choice :
See ...
- 41
2
votes
0
answers
123
views
Constructing a lattice out of the set of metrics
Let $X$ be a space, and $d_1$ and $d_2$ be two metrics on $X$.
Define $S(x,y)= ${$\Sigma_2^l Min${$d_1(x_{k-1},x_k),d_2(x_{k-1},x_k)$}$:x_1=x, x_l=y, l finite $} $x$ and $y$ are two points in $X$
...
- 31
9
votes
4
answers
2k
views
Triangulating hypercubes
Motivation: I'm working on a computational problem at the moment, and have some very good routines for natively working with simplicial complexes and calculating homology, but the structures I'm ...
- 387
18
votes
7
answers
2k
views
Superfluous definitions
It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.
For if a and b are elements of R, and writing + for the group operation then applying ...
16
votes
2
answers
4k
views
Is there a "disjoint union" sigma algebra?
I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally:
For an indexed family of sets $\{A_i\...
- 629
14
votes
4
answers
1k
views
Localic locales? Towards very pointless spaces by iterated internalization.
One can think of locales as (generalizations of) topological spaces which don't necessary have (enough) points. Of course when one studies locales, one "actually" studies frames,
certain sorts of ...
- 17.6k
7
votes
0
answers
310
views
The self-duality of topological compactness
The impatient reader can skip my attempt at motivation and go straight my "Question formulations for the impatient."
In a failed(?) attempt at discovering something new, some years ago I ...
- 17.6k
1
vote
1
answer
606
views
About deformation retract
Let $X\subset Y$ be CW-complexes. Denote $i\colon X\to Y$ be an inclusion map.
Is it true that $i$ is deformation retract if and only if $i$ is homotopy equivalence?
When I saw some papers about h-...
- 13
7
votes
2
answers
594
views
Computational cost of converting between 3-manifold presentations
Given a 3-manifold presented as a triangulation, a Heegaard splitting, or a Dehn surgery, what is the computational cost of converting to the other two presentations? I would like Heegaard splittings ...
- 135
37
votes
5
answers
7k
views
Example of sequences with different limits for two norms
I was explaining to my students that if there is an inequality between two norms, then there is an inclusion between their spaces of convergent sequences, with matching limits. I then proceeded to ...
- 2,054
37
votes
5
answers
5k
views
Locales and Topology.
As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
5
votes
1
answer
293
views
semigroups acting as continuous functions on regular rooted trees
Let $T$ be a regular rooted tree. Make $T$ into a metric space by making each edge isometric to the unit interval. What is known about what semigroups can act as continuous functions on $T$ such ...
- 51
5
votes
2
answers
655
views
$C^n$ And Forcing: Reading a Recent Paper By Kunen
While reading a recent paper by Kunen arxiv.org/abs/0912.3733, which deals with PFA and the existence of certain differentiable functions, (defined on all of $\mathbb{R}$) which map certain $\aleph_1$-...
- 1,615
17
votes
3
answers
3k
views
Is there a way to graphically imagine smash product of two topological spaces?
Recently I've been reading "Topology" by Klaus Janich. I find this book very entertaining as it contains lots of graphical illustrations that appeal to my "geometrical" imagination. In paragraph 3.6 ...
- 965