Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,602 questions
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closed meagre sets
A closed meagre subset of $[0,1]$ is either countable or homeomorphic to the Cantor set: either way it is $0$-dimensional.
Q.1. Is every closed meagre subset of an $n$-dimensional locally compact ...
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When LCS is isomorphic to subspace of some function space?
Updated: Following Michael's suggestion, I rephrase the question slightly.
Given a locally convex (Hausdorff) topological vector space (LCTVS), when is it isomorphic to a subspace of some function ...
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Spectral sequences in Hypercohomology of sheaves (For a complex of acyclic sheaves) - Follow-up to previous question
Alright, this is a follow-up to my previous question (Spectral sequences in Hypercohomology of sheaves), sorry I took so long to reply. Let $X$ be a topological space, let $F^\bullet$ be a cochain ...
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Metric on the set of Polyhedral Decompositions of a Compact Metric Space
Let $(X,d)$ be a compact metric space of finite diameter. Recall that we can always compute the Hausdorff distance between subsets $A, B \subset X$ via
$$ d_H(A,B) = \max\left[\sup_{a\in A}\inf_{b\in ...
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properties of $\beta\omega\setminus\omega$ minus the P-points
Let $X=\beta\omega\setminus\omega$ and let $Y=X\setminus P$ where $P$ is the set of P-points in $X$. Then $P$ is dense in $X$ if we assume CH but $P$ may be empty otherwise (Shelah). In the one case $...
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Counterexemple to Urysohn's lemma in a topos without denombrable choice ?
Hello !
The Urysohn's Lemma assert that in every topological spaces which is normal two closed subset may be separated by a real valued function. It's proof use axiom of countable choice (but not the ...
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Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.
Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$
Let us define the following:
Condition 1: $X$ can be covered by ...
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Relation on the set of connected components of the $\mathbb{C^*}$-fixed points locus coming from the Bialynicki-Birula decomposition
Let $X$ be a smooth variety with an action of $\mathbb{C}^*.$ One has the so-called Bialynicki-Birula decomposition of $X$ given by stable manifolds: $$X=\bigcup_N X^+(N),$$ where $N$ varies in the ...
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Connected level sets
This may be an ill-posed question, but suppose I have a collection of continuous, bounded, scalar-valued nonnegative functions $f_1(x,y),\dots,f_n(x.y)$ defined on the closed unit disk. Given a ...
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Spectral sequences in Hypercohomology of sheaves
Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this ...
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Trivial cobordism group in dimensions 1, 3, 7 related to H-space structures on the spheres in these dimensions?
Is there a connection between the existence of H-space structures on $S^1$, $S^3$ and $S^7$ and the fact that every (closed) 1-manifold, 3-manifold and oriented 7-manifold is a boundary, or is this a ...
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Where else do the (topology) separation axioms turn up?
As an undergraduate I learned point-set topology from Munkres's book, as did many others.
One topic that gets a lot of attention is the separation axioms. For example, a space $X$ is normal if any ...
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Partitions of an interval
This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" ...
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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 ...
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Does every ultrafilter has single limit imply Hausdorff separation
If a topological space $X$ enjoys the property that every ultrafilter $U$ on $X$ has a single limit, must $X$ be a Hausdorff space?
(Ultrafilters here consist of arbitrary subsets (so not necessarily,...
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topological equivalence of ODEs
Let $n$ be a non-negative integer. $\;\;$ Let $\: f : \mathbb{R}^n \to \mathbb{R}^n \:$ and $\: g : \mathbb{R}^n \to \mathbb{R}^n \:$ be Lipschitz.
Define the relation $\stackrel{f}{\sim}$ on $\...
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union of Stone-Cech remainders
Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
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Condition to ensure that the product of closed maps be closed
If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map
$$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots \times Y_n : (x_1, \...
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Number of non-intersecting non-homotopic simple closed curve
How many simple closed curves can be put on a orientable genus $g$ surface $\Sigma_g$ such that the following are true:
The curves are pairwise non-homotopic
The curves are pairwise set-theoretically ...
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Do homotopic non-intersecting simple closed curves separate the surface?
Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
They are homotopic to each other.
They are set-theoretically disjoint.
Is $S\setminus(C_1 \cup C_2)$ ...
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Topology of the "normal spectrum" of a commutative von Neumann algebra
Kadison and Ringrose define normal states $\omega$ of a von Neumann algebra $A$ as such that $\omega(H_\alpha)\to \omega(H)$ for each monotone increasing net of operators $H_\alpha$ with least upper ...
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Hypercohomology of a complex of sheaves that might be acyclic (or might not)
Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves
...
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Is there a natural topology on the set of open sets ?
Given a topological space $(X,\mathcal{O})$ can one assign a natural topology to $\mathcal{O}$ such that
1) The intersection of a compact set of open sets is again open,
2) The maps $\cap,\cup:\...
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Borel functions on $\omega_1$
Endow $\omega_1$ with order topology. It is easy to show that each continuous function $f\colon \omega_1\to \mathbb{R}$ is eventually constant. Is the same true for Borel functions?
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Graphs, non-Hausdorfness and Wallman compactifications of non-regular spaces
Given a non-Hausdorff space $X$, one can form a graph $G_X$: vertices the points of $X$, edges indicating point pairs not separated by open sets. Up to graph-theoretically (but not topologically) ...
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General topology terminology questions
In a Hausdorff but not regular space, collapsing certain closed sets to a point may produce a non-Hausdorff space. Does there exist a term for closed sets one may collapse and still have a Hausdorff ...
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Square of a continuous map
Recently a student asked me the following (elementary looking) question :
If $T$ is an invertible linear transformation of some finite-dimensional space $E$ into itself which factorizes as $T = f \...
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Which spaces are characterized by functions with compact support ?
It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic.
Is there a class $\...
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closed set and z-ultrafilter on normal space
Let $X$ be a completely regular, Hausdorff topological space and let $\cal F$ be a $z$-ultrafilter on $X$. Then for each zero set $W$ in $X$, either $W\in \cal F$ or there exists $Z\in \cal F$ such ...
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Infinite closed partition of the real line with no closed infinite unions
Is there a partition of the real line into infinitely many closed subsets so that no infinite union of these subsets (except the whole space) is closed?
This question was asked also at math....
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Metrizable implies hemicompact
In the paper
R. Arens: A Topology for Spaces of Transformations, Ann. of Math. 47(1946), 480-495
the author states in the introduction that if $B$ is a metric space and the space of continuous ...
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Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
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Stone-Čech compactification of $\mathbb R$
Let $\beta X$ - is a Stone-Čech compactification of $X$. $I=(-1,1)$ - is an interval of the real line. Is it true that $\beta \mathbb R\setminus I = \beta(\mathbb R\setminus I)$? In other words, it ...
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Least cardinality of a set of points in the plane
What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...
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Why are profinite topologies important?
I hope this is not too vague of a question. Stone duality implies that the category Pro(FinSet) is equivalent to the category of Stone spaces (compact, Hausdorff, totally disconnected, topological ...
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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, ...
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Moore path space.
Let $X$ a topological space and $MX$ the Moore path space of $X$
there is two maps from $\alpha,\omega: MX\rightarrow X$ (evaluation in 0 and evaluation at the total length).
The classical path ...
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Topology on Set of Prime Filters of a Distributive Lattice
Given a distributive lattice $A$ we can look at $Spec(A)$, whose points are prime ideals and its open sets are given similarly to the Zariski topology on Spec of a ring. That is, the basis of open ...
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Possible homogeneity of infinite dimensional Sierpinski carpet analogues?
Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion.
Construct subsets $S_n$ by removing points from $H$ if for any $m$,
at least $n$ of the ...
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When is a Topological pushout also a Smooth pushout?
I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:
Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...
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A closure operation on subsets of ${\Bbb Z}[x]$
Given a(n infinite) set $S\subset {\Bbb Z}[x]$ (integer polynomials), write $R_S$ for the topological closure of the set of all complex roots of all $p\in S$. Then write $\hat{S}$ for the set of all ...
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(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 ...
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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 ...
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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 (...
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A boundary-preserving map on the unit disk
We are given a (closed) ball $D^n$ and a (continuous) map $f: D^n \to D^n$, that is identity on the boundary of $D^n$.
Let $C$ be a subset of $D^n$, and let $f^{-1}(C)$ be the inverse image of $C$ ...
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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, ...
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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 ...
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Example of a weak Hausdorff space that is not Hausdorff?
I've looked on the web and haven't found a simple example.
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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 ...
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When is a Homology Class Represented by a Submanifold? [duplicate]
Possible Duplicate:
Cohomology and fundamental classes
Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi_*[N]\in H_*(M)$ ...
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