All Questions
5,184 questions
1
vote
2
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360
views
Is this quotient space of Q_p contractible?
Let $X_{p} = \mathbb{Q}_{p} / \sim $, where $\sim$ is defined by:
$x\sim 0 \Leftrightarrow x\in \mathbb{Q}$
$X_{p}$ is path-connected, because (unless I'm making some horrible mistake,) for any $x\...
- 713
2
votes
1
answer
743
views
weak metric space
In the definition of a metric space, replace the triangle inequality by the weaker inequality
d (x, z) ≤ C max {d (x, y), d (y, z)},
where C is a positive constant (depending on the "metric", ...
53
votes
4
answers
24k
views
When is $L^2(X)$ separable?
I have never studied any measure theory, so apologise in advance, if my question is easy:
Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable?
In reality, I am interested in ...
- 12.3k
20
votes
2
answers
1k
views
Rugged manifold
It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary.
Is it still true for topological manifold?
Comments:
To proof the smooth ...
- 45k
38
votes
5
answers
5k
views
Does "compact iff projections are closed" require some form of choice?
There are many equivalent ways of defining the notion of compact space, but some require some kind of choice principle to prove their equivalence. For example, a classical result is that for $X$ to be ...
- 53.3k
5
votes
3
answers
2k
views
Discrete subspaces of Hausdorff spaces
does every infinite hausdorff space contains a countable infinite discrete subspace?
- 531
2
votes
2
answers
657
views
Varieties, Frechet Completions, and Regular Functions
Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local ...
- 1,462
4
votes
0
answers
939
views
Proofs of Baire category theorem
I would like to have a list of proofs of the fact that the real line is not meager (also very useful would be a reference to such a list, if it already exists somewhere).
My motivation is the ...
- 315
3
votes
0
answers
294
views
Monomorphisms in geometry
What is known about monomorphisms in the following categories:
Schemes
Complex manifolds
$C^\infty$--manifolds
and any other kinds of geometric objects that you might think of.
How do we choose a ...
- 123
5
votes
3
answers
551
views
Nonmetrizable uniformities with metrizable topologies
I'm looking for such pathological examples of uniform spaces which are not metrizable, but whose underlying topology is metrizable. Willard in his General Topology text constructs such a uniformity ...
- 3,004
2
votes
0
answers
77
views
Characterizing local homeomorphisms into an exponent
Let $X$,$Y$, and $Z$ be (compactly generated) spaces. Suppose $f:Z \to Y^X$ is a local homeomorphism. How can we tell this from its adjoint $\tilde f:Z \times X \to Y$? I.e., I want a property $P$ ...
- 15.5k
11
votes
1
answer
536
views
Can dividing out a group action can increase the Lebesgue dimension ?
Given any space $X$ of Lebesgue dimension at most $n$. Suppose a group $G$ acts on $X$ continuously. Can the dimension of the quotient $G\backslash X$ exceed the dimension of $X$?
I know examples, ...
- 11.1k
3
votes
1
answer
4k
views
A sequence with no convergent subsequence without choice
By Tychonoff Theorem $\prod_{\mathbb R} [0,1]$ is compact and since $\mathbb R=2^{\omega}$, if for $\alpha \in 2^{\omega}$, $x_n(\alpha)=\alpha(n)$ then if we consider a subsequence $x_{n_0}, x_{n_1}, ...
- 3,804
8
votes
2
answers
875
views
Is the mapping cylinder of a Serre fibration also a Serre fibration?
If we have a Serre fibration $p: E \rightarrow B$ with fiber of homotopy type $S^{k-1}$, then we can create a fibration with contractible fiber by first taking the mapping cylinder $M_p$ of $p$ to get ...
- 1,207
1
vote
2
answers
378
views
Is this a pre-ordered commutative semigroup?
Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the ...
- 33.3k
40
votes
1
answer
3k
views
Is every connected scheme path connected?
Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following.
Let's ...
- 47.5k
3
votes
1
answer
645
views
Is the Hopf link a Brunnian link?
I'm trying to fill a woeful gap in my topological knowledge and learn a little knot and link theory (I'll be recording my progress on the nLab, starting with a page on links). Not wishing to write ...
- 26.8k
11
votes
2
answers
2k
views
Hausdorff dimension of the boundary of an open set in the Euclidean space - lower bound
I consider a bounded open set $A$ in ${\mathbb R}^d$. Is the Hausdorff dimension of the boundary of $A$ at least $d-1$ ? I thought I would have found a result on this problem in any textbook about ...
- 631
11
votes
9
answers
1k
views
Proving the impossibility of an embedding of categories
A number of topological invariants take the form of functors $\mathscr{T}\to\mathscr{G}$, where $\mathscr{T}$ is the category of all topological spaces and continuous functions, and $\mathscr{G}$ is ...
- 5,759
7
votes
0
answers
433
views
Ever seen a ringed group?
A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
- 12.3k
19
votes
2
answers
2k
views
Complete metric on the space of Jordan curves?
I was interested in putting a complete metric on the space of Jordan curves. Say, just planar Jordan curves contained in $B(\bar{0}, 2) \backslash B(\bar{0}, 1)$ which separates $\bar{0}$ and infinity....
- 375
4
votes
1
answer
511
views
A question about open subsets of Hilbert space
If H is (a separable and infinite dimensional) Hilbert space and if U is a non-empty
open subset of H that is not connected, does the boundary B of U always have at least
one component that is not a ...
- 6,215
4
votes
2
answers
364
views
General linear inverse monoid
Let $V$ be a finite dimensional vector space over some field (say, $\mathbb C$). Consider the set $\operatorname{GLI}(V)$ of all linear isomorphisms between subspaces of $V$. This is a monoid under ...
user6976
7
votes
1
answer
1k
views
Minimal conditions for the exponential law for compact-open topologies
What are the minimal conditions on three topological spaces $X,Y$ and $Z$ such that with the compact-open topology the map
$${(X^Y)}^Z \to X^{Y \times Z}$$
given by taking adjoints is a homeomorpism....
- 891
2
votes
2
answers
390
views
Is a compactly generated Hausdorff space functionally Hausdorff?
Question is the title. I suspect the answer is no, without some further conditions (clearly, normal is sufficient). Pointers to counterexamples would be appreciated, but not necessary.
- 35.5k
6
votes
0
answers
510
views
The Mapping Cylinder of a Pullback Square
Suppose I have a pullback square, which I think of as a map from the fibration
$q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$
from the mapping cylinder $M$ of $X\...
- 12.5k
27
votes
1
answer
4k
views
connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 6,224
4
votes
1
answer
490
views
A question on PL-topology and polytopal complex
Question : $C$ is a pure, full-dimensional polytopal complex(a special case of a regular cell complex) in $\mathbb{R}^d$. I know that the boundary of the underlying set is a PL-sphere. Is it true that ...
- 113
2
votes
0
answers
185
views
Simple topological question on taking complements inside a simplex
We would like to know if the following claim is true:
(If you don't know the definition of a tropical hyperplane, then please consider the case when d=3)
Let $P_1,\cdots,P_d$ be full dimensional (...
- 113
3
votes
1
answer
376
views
Chaos in uniform spaces
Let $Dom$ be a uniform space, and $\hspace{.04 in}f$ be a continuous function from $Dom$ to itself satisfying:
For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural
number $n$ ...
user5810
9
votes
2
answers
3k
views
Compact Hausdorff spaces without isolated points in ZF
$S$ is uncountable := $\vert\mathbb{N}\vert<\vert S\vert$
$S$ is noncountable := $\vert S\vert \not\leq \vert\mathbb{N}\vert$
$(X,T)$ is a nice space := $(X,T)$ is a compact Hausdorff space ...
user5810
11
votes
3
answers
1k
views
Can there be two continuous real-valued functions such that at least one has rational values for all x?
Of course, no continuous real valued non-constant function can attain only rational or irrational values, but can there be a pair of nowhere-constant continuous functions f and g such that for all x, ...
- 656
15
votes
5
answers
1k
views
Monoids with infinite products
Say a monoid $M$ has infinite products if, for any (possibly infinite) sequence $(m_1,m_2,\ldots)$ of elements of $M$, there exists an element $m_1m_2\cdots\in M$, satisfying some good properties. ...
- 8,659
5
votes
1
answer
301
views
Lebesgue dimension of closures satisfying the Z-set condition
Given any subspace $A\subset X$ of a topological space with Lebesgue dimension $\le N$.
Let $\bar{A}$ denote the closure of $A$. Assume, that the pair $(\bar{A},A)$ satisfies the Z-set condition, i....
- 11.1k
5
votes
2
answers
3k
views
Closedness of finite-dimensional subspaces
Is the (algebraic) span a finite set of vectors in a Hausdorff topological vector space over a complete field always closed?
I suspect yes, but I can't come up with a proof, and it seems like locally ...
user5810
5
votes
2
answers
310
views
A question about homeomorphic subsets of Hilbert space
Let H be a an infinite dimensional and separable Hilbert space. Let C be a closed and
bounded subset of H that is not compact. Does there always exist a closed and unbounded
subset of H which is ...
- 6,215
15
votes
0
answers
2k
views
Covers of $Z^k$
This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...
11
votes
2
answers
843
views
covers of $Z^\infty$
Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of ...
user6976
4
votes
4
answers
599
views
A question about indecomposable continua.
The term "continuum" is often used to mean a compact and connected metric space. But it is
also used in a broader sense to mean any infinite, complete, separable and connected metric
space-which is ...
- 6,215
24
votes
5
answers
8k
views
totally disconnected and zero-dimensional spaces
When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: covering dimension, small ...
- 3,605
8
votes
0
answers
302
views
In a locally contractible space can we find local bases of contractible sets whose closures are locally contractible?
In a locally contractible topological space $X$ is it possible at each point $x$ to find a local basis of contractible sets $U_i\ni x$ such that the closure of each set $\overline{U_i} \subset X$ is ...
7
votes
3
answers
590
views
Expressing any f(x,y) using only addition and unary functions?
Suppose we have a continuous function $f:R^2\rightarrow R$. I was told of the following remarkable theorem: $f$ can be expressed as the composition of continuous unary functions (that is, functions ...
- 3,979
1
vote
1
answer
908
views
What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...
10
votes
1
answer
2k
views
Different forms of compactness and their relation
Given a topological space X one can define several notion of compactness:
X is compact if every open cover has a finite subcover.
X is sequentially compact if every sequence has a convergent ...
- 3,004
0
votes
1
answer
423
views
What Is This Quotient Space?
Let $X$ be a finite CW-complex with only even cells $x_1,\ldots, x_k$ and let $Y$ be the complex obtained by attaching one more even cell to $X$, call it $y$. Assume both $X$ and $Y$ are connected. ...
- 61
3
votes
4
answers
829
views
Does countable compactness imply local compactness in Hausdorff spaces?
The question arose while comparing the notions of compactness, countable compactness, local compactness, and "Lindelofness" in Hausdorff spaces. It is straightforward to show that compactness implies ...
- 544
78
votes
12
answers
12k
views
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
- 2,392
4
votes
2
answers
544
views
Membership problem in monoids
What is the simplest example of a monoid with undecidable membership problem? In other words, I'm looking for a concrete monoid $S$ such that there is no algorithm which takes elements $s_1,...,s_n$ ...
- 41
11
votes
2
answers
1k
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Normal Varieties
Let X be a complex normal variety and U a subvariety that is open in the analytic topology. Then the map $\pi_1(U) \to \pi_1(X)$ coming from the map $U \subset V$ is surjective - why is this?
edited ...
- 255
24
votes
6
answers
5k
views
A good place to read about uniform spaces
I'd like to learn a bit about uniform spaces, why are they useful, how do they arise, what do they generalize, etc., without getting away from the context of general topology. I have to prepare an ...
- 3,004