Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,601 questions
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Is the space of continuous functions from a compact metric space into a Polish space Polish?
Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup_{k\in K}\ ...
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When is the group of homeomorphisms of a compact space locally compact?
When is the group of homeomorphisms of
a compact space locally compact?
I am interested in finding out when the group of homeomorphisms of a compact topological space $X$ (with appropriate topology ...
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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" ...
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Why free topological groups on Tychonoff spaces?
This is a question of the motivation for a common assumption found in the literature.
The free topological group $F(X)$ on a space $X$ exists for all spaces $X$ (It seems this was first shown by ...
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Topological degree theory
Let $D$ be a region in $R^n$. If $f:D\to R^n$ is continuous, nonzero on $\partial D$ and of Brower degree 0, does there exists a continuous function $g=f$ on $\partial D$ and $g\neq 0$ on $D$?
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Quotients of topological groupoids
The issues that arise when moving from topological groups to topological groupoids are (at least to me) both subtle and interesting. Recently, I was reading a paper of R. Brown and J.P.L. Hardy from ...
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When do maps of ineffective orbifolds descend to their effective part?
If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...
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Profinite topologies
We can define two topologies on a group $G$ by considering all normal subgroups of finite index (resp. of index a finite power of $p$ - where $p$ is a prime) as basis of $1\in G$.
My questions: Under ...
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Closed connected subset of a connected set
Let $A$ be a closed set and let $B$ be a connected set such that $A \subset B$.
Does there always exist a closed connected subset $C$ of $B$ that contains $A$?
What if $B$ is path connected, is ...
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What abstract nonsense is necessary to say the word "submersion"?
This question is closely related to these two, but the former doesn't go far enough and the latter didn't attract much attention, and anyway I want to ask the question slightly differently.
Recall ...
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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 ...
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Abelian groups as fundamental groups of topological groups
Hi,
It is well known that the fundamental group of a topological group is abelian, and that every group is the fundamental group of some topological space.
My question is: Does every abelian group ...
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The Space of Cellular Maps
Let $X$ and $Y$ be CW complexes.
Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...
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Locally compact separable metric spaces
Hi,
Is it true that for every locally compact separable metric space $E$ there exists a sequence $(K_n)_{n\in\mathbb{N}}$ of compact subsets of $E$ such that $K_n\subset\stackrel{\circ}{K_{n+1}}$ and $...
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Which properties of finite simplicial sets can be computed?
A simplicial set $X$ is a a combinatorial model for a topological space $|X|$, its realization, and conversely every topological space is weakly equivalent to such a realization of a simplicial set. I ...
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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 ...
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Homotopy theory for spanning trees of a graph
I am studying a paper of L. Lovász, ``A homology theory for spanning trees of a graph,'' but professor Babai has told me that Lovász later realized that this work is better framed in the language of ...
- 8,803
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Topological dimension, is it local?
Let $n\in\mathbb N$ and $X$ be a complete metric space.
Assume that there is $\epsilon>0$ such that
$$\dim B_\epsilon(x)\le n$$
for any $x\in X$.
Is it true that $\dim X\le n$?
Here $\...
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local structure of free $\mathbb{R}$ actions
Assume the topological group $\mathbb{R}$ acts properly on a space $X$. Does then the projection map $p:X\rightarrow \mathbb{R}\backslash X$ have local sections ?
(for every $\mathbb{R}x\in \mathbb{R}...
- 15.3k
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chains and countability
Given a point $x$ in a topological space $X$. I was wondering, whether one can always find a local basis at $x$, which is totally ordered (a chain) under inclusion. For example this is true for spaces,...
- 236k
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A property of continuous maps with respect to compact subsets
I'm interested in continuous maps between topological spaces $f:X\to Y$ such that for any compact subset $L$ of $Y$ contained in $f(X)$, there is a compact subset $K$ of $X$ such that $L$ is contained ...
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An ultrafilter and a partition
Let $S$ is a partition of a set $U$. Let $c$ is an ultrafilter on $U$.
Prove or disprove this conjecture:
At least one of the following is true:
$\exists D\in S, C\in c:C\subseteq D$
or
$\exists C\...
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Hilbert space automorphisms realized as induced by transformations of some base-spaces
Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
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End point compactification for metric spaces
Freundenthal introduced ends of topological spaces and the end point compactification of locally compact topological spaces adding one point for each end of the topological space (see here).
For ...
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Discriminant locus in knot space
Consider the space $K$ of all immersions of $S^1$ into $\mathbb R^3$.
The set of knots with self-intersection is a discriminant in $K$ and divide it into "chambers".
Let $f$ be a knot with $n$ double ...
- 44.4k
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pseudo-Anosov maps on surfaces with boundary
In "Automorphisms of Surfaces after Nielsen & Thurston" by Casson & Bleiler (on pages 75 - 80) they discuss classifying automorphisms of a surface. They show that, if $S$ is a closed ...
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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 ...
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Topological dimension versus cohomological dimension
This should be really well known but I don't seem to find a statement about it nor a question in MO answering this.
Consider a Compact Hausdorff topological space $X$. The cohomological dimension of ...
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Isocontours of depth and magnitude of gradient
We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...
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Are coordinate functions on topological vector spaces always continuous?
Let $V$ be a Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ such that
$\;$ for all functions $c : B\to \mathbb{K}$, if $\displaystyle\sum_{...
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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\...
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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", ...
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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 ...
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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 ...
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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, ...
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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 $P$ ...
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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}, ...
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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Why are topological ideas so important in arithmetic?
For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in ...
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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 ...
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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
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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 ...
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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 (...
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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, ...
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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 ...
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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 ...
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