All Questions
5,184 questions
19
votes
4
answers
4k
views
When is a finite cw-complex a compact topological manifold?
I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
- 732
0
votes
0
answers
559
views
Visualizing self-homeomorphism of a cylinder over a torus
A cylinder over a torus is by definition $S^1 \times S^1 \times I$ , here $I=[0,1]$.
One way to visualize it is to thick a torus in $\mathbb{R}^3$. ( $S^1 \times I$ is an annulus, and revolve it (...
- 93
5
votes
1
answer
381
views
Ring of a Spectral Space
It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under ...
- 10.4k
2
votes
1
answer
430
views
Automorphism of first homology and mapping class group
It is known that for a torus $\Sigma$, every automorphism of $H_1(\Sigma; \mathbb{Z})$ is induce by an orientation preserving self-homeomorphism of $\Sigma$ unique up to isotopy. In onther words, ...
- 93
1
vote
1
answer
317
views
Mapping class group and cylindrical structure
Let us fix a torus $\Sigma=S^1 \times S^1$. We consider a cylinder $\Sigma \times I$ and a data $(\Sigma\times I, \Sigma\times 0, \Sigma\times 1, f_{0},f_{1})$. Here $f_{i}$, called parametrization, ...
- 93
18
votes
2
answers
3k
views
Example of a weak Hausdorff space that is not Hausdorff?
I've looked on the web and haven't found a simple example.
- 728
11
votes
5
answers
5k
views
A criterion for the sum of two closed sets to be closed ?
Let $V$ and $I$ be two closed subsets of a Banach space $A$.
The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$.
I would like to know whether $I+V$ ...
- 452
2
votes
0
answers
369
views
Constructing the Stone space of a distributive lattice
Does anyone have a good reference for the method of giving a topology to a distributive lattice as outlined in M.H. Stone's "Topological representation of distributive lattices and Brouwerian ...
- 10.4k
1
vote
1
answer
2k
views
Covering the Rationals -- A Paradox? [closed]
Covering the Rationals -- A Paradox?
The following seems to yield a paradox. Can anyone provide the proper resolution?
Preamble
It is easy to show that between any two reals there is a rational. If ...
113
votes
4
answers
13k
views
Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
- 22.1k
1
vote
3
answers
585
views
Terminology for certain monoids which are to monoids like fields are to rings
Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "...
10
votes
1
answer
869
views
Completeness of Borel measure
Let $X$ be a compact Hausdorff space and $\mu$ a finite Borel measure without atoms which is outer regular with respect to open sets and inner regular with respect to compact sets. Can such measure be ...
- 277
11
votes
3
answers
1k
views
Is the reals the smallest connected ordered topological ring?
The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to ...
user2529
5
votes
1
answer
738
views
Characteristic classes of a fibered sum
I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general.
Let $M_1$ and $M_2$ be smooth $n$-manifolds (maybe closed, for simplicity),...
- 732
92
votes
3
answers
14k
views
Is every sigma-algebra the Borel algebra of a topology?
This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some ...
- 236k
46
votes
2
answers
5k
views
Continuous bijections vs. Homeomorphisms
This is motivated by an old question of Henno Brandsma.
Two topological spaces $X$ and $Y$ are said to be bijectively related, if there exist continuous bijections $f:X \to Y$ and $g:Y \to X$. Let´s ...
- 11.5k
105
votes
5
answers
16k
views
Independent evidence for the classification of topological 4-manifolds?
Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? ...
- 2,337
-1
votes
1
answer
542
views
Fuzzy topology : references [closed]
Hey. I'm looking for references in fuzzy topology. Does anyone know a good book ?
- 11
5
votes
3
answers
676
views
Does every compact Hausdorff ring admit a decomposition into primitive idempotents?
Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in $\mathbf{R}$)...
- 1,498
1
vote
0
answers
202
views
Soft sheaves on indiscrete paracompact spaces
Let $X$ be some space, I have basically 2 questions:
1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
- 417
14
votes
2
answers
2k
views
Well-pointed space which is not locally contractible
I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...
- 6,210
6
votes
0
answers
561
views
Continuous images of Cantor cubes
The original title of this question was "Is there only one (up to homeomorphism) zero-dimensional homogeneous dyadic space of weight $\mathfrak{c}$?". I changed it with the hope of getting a bit more ...
- 11.5k
2
votes
0
answers
140
views
Products for probability theory using zero sets instead of open sets
(For all of this post, at least Countable Choice is assumed to hold.)
For all Tychonoff spaces $\langle X,\mathcal{T}\hspace{.06 in}\rangle$ :
Define $\mathbf{Z}(\langle X,\mathcal{T}\hspace{.06 in}\...
user5810
1
vote
2
answers
1k
views
Possible errata in Nicolas Bourbaki's General Topology -I, Chapter 1 Exercise 2 ?
Here is the text of Exercise:
2 a) Let $X$ be an ordered set. Show that the set of intervals
$\left[x, \rightarrow\right[$ (resp. $\left]\leftarrow, x\right]$)
is a base of topology on $X$; ...
- 163
2
votes
3
answers
614
views
A simple question on the closure of the image of a morphism
Let $X$ be a complex irreducible quasi-projective variety, $f:X\longrightarrow\mathbb{P}^N$ a morphism, $H\subset\mathbb{P}^N$ a hyperplane, $Z:=f^{-1}(H)$ which is irreducible, $Y\subset X$ a ...
- 1,159
3
votes
2
answers
715
views
Separation axioms
Reading about separation axioms, I wonder:
Is there a separation axiom weaker than $T_2$ but stronger than $T_1$? $T_{1.5}$?
I suppose there are some separation axioms stronger that $T_6$, how many ...
- 531
6
votes
5
answers
919
views
A question about local connectedness in metric spaces
Must every compact and connected metric space be locally connected at at least one
of its points?
- 6,215
11
votes
2
answers
2k
views
Two Definitions of "Character" of topological groups
When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows:
Let $G$ be a topological group. A ...
- 945
4
votes
2
answers
881
views
extracting a convergence subnet from a sequence which is Cauchy on every bounded subset of $\mathbb N$.
I have a certain sequence $x_n$ in a complete and bounded metric space and I would like to prove that it has a convergent subnet (not necessarily subsequence). The best that I was able to do, until ...
- 6,034
6
votes
1
answer
873
views
Products with compactly generated spaces
It is well known that if $X$ and $Y$ are topological spaces with $X$ locally compact Hausdorff and $Y$ compactly generated, then $X \times Y$ (with the ordinary product topology) is compactly ...
- 15.5k
2
votes
1
answer
341
views
showing uniformly continuous
Let $(X,d)$ be a metric space and $(a_n)$ be a sequence of distinct points in $ X$ such that each $a_n$ is a limit point of $X$. If $U_n$ 's are mutually disjoint open neighbourhoods of $a_n$ in $X$. ...
- 21
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.
- 127
3
votes
1
answer
737
views
Klein Bottle exception to the Heawood Conjecture [duplicate]
Possible Duplicate:
The Klein bottle and the Heawood Conjecture
It is well known that the Heawood Conjecture states that the bound for the number of colours which are sufficient to colour a map ...
- 1,441
7
votes
0
answers
624
views
"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
- 18.7k
2
votes
1
answer
220
views
Extending BAs to weakly countably distributive algebras.
Suppose $\mathscr{A}$ is a complete Boolean algebra (assume it is c.c.c. if you wish). Is there any, say, canonical embedding $\mathscr{A}\subseteq \mathscr{B}$ into a complete Boolean algebra which ...
- 55
3
votes
2
answers
384
views
ED compact $K$ such that $C(K)$ is not a dual Banach space
Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not hyperstonean). ...
- 55
5
votes
0
answers
263
views
Coloring $\mathbb{Z}^k$ and a fixed point theorem
This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in ...
user6976
12
votes
1
answer
1k
views
(Closures of sets of) operations in topological groups.
Let $G$ be a topological group. For each $n \in \mathbb{Z}$, consider the continuous functions $f_{n} \colon G \to G : x \mapsto x^{n}$, and set $F := \{f_{n} \mid n \in \mathbb{Z}\}$.
Is there a ...
- 1,498
2
votes
1
answer
232
views
Could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology?
Let $X$ be a Tychonof space and $\beta X$ is its compactification. Then could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology?
- 654
4
votes
2
answers
2k
views
Are countable unions of metrizable spaces metrizable too?
Suppose that $X=\bigcup_{n=1}^\infty K_n$ is a topological space, $K_n$ is a metrizable subspace in $X$ for every $n \in \omega$, then $X$ is a metrizable space?
In metrizable spaces, compactness is ...
- 654
8
votes
1
answer
446
views
Topological conditions forcing continuity
Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous.
Question: Under what ...
- 11.8k
0
votes
2
answers
359
views
some questions on Lindelöf property
I have several questions on Lindelöf property.
If every point countable open cover of $X$ has a countable subcover (Condition A), does $X$ have Lindelöf property? How far is having Condition A from ...
- 654
4
votes
3
answers
515
views
Does every Lindelof uniform space have a Lindelof completion?
Recall Lindelof = every open cover has a countable subcover. Is the question of the title answered by known material from some standard text on uniform spaces?
Note it is well known to be true for ...
- 1,704
1
vote
0
answers
430
views
Universal Hausdorff Space [duplicate]
Possible Duplicate:
Largest Hausdorff quotient
Is there a left adjoint to ${\mathbf{Haus}}\to{\mathbf{Top}}$? Here ${\mathbf{Haus}}$ is the full subcategory of Hausdorff spaces in ${\mathbf{Top}}$...
user2529
9
votes
0
answers
741
views
Pseudocycle definition of open Gromov-Witten invariants
I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as long!...
- 1,129
14
votes
2
answers
1k
views
Economical hard word problem
Can anyone give me an example of a very simple word problem, where by "simple" I mean that it has very few generators and relations, that is nevertheless insoluble. To make the question easier, I am ...
- 29k
22
votes
2
answers
1k
views
Toposes (topoi) as classifying toposes of groupoids
A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Butz and Moerdijk have shown that if the topos has ...
- 38.6k
6
votes
3
answers
679
views
Questions on 3-manifolds with a given boundary
I have the following question:
For a given two-dimensional Riemann surface $C$,
Is there a way to classify all topologically distinct
three-dimensional compact manifolds $M$ whose boundary is $C$,
i....
- 424
6
votes
2
answers
1k
views
On the uncountability of zero sets
If $f$ is any real-valued function, we define its zero set $Z_f = \{ x : f(x) = 0 \}$. Obviously, the zero set of a nice function can be uncountable. e.g., if $f(x) = 0$ on an uncountable domain.
I ...
- 8,512
23
votes
12
answers
18k
views
A book in topology
I will have to teach a topology course:
it starts in point set topology and ends at fundamental group of $S^1$.
In the past I have used two different books:
Elementary Topology. Textbook in Problems,...