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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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 $\mathbf{R}$)...
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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}\...
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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 ...
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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 ...
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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 $X$. ...
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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 ...
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"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 ...
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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 ...
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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?
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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 ...
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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). ...
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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 ...
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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 ...
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Question about analytic curves
Here a question that has me stumped. Maybe someone familiar with algebraic or differential curves can help. Suppose that $\gamma:[0,1] \rightarrow \mathbb{C}$ is an analytic function. Is it true ...
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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!...
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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 ${\mathbf{Top}}$...
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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....
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Applications of Eckmann-Hilton argument to topology
There have been a couple of posts and questions on MathOverflow about the proofs of the following two facts:
Fact 1: if $X$ is a topological space, then $\pi_k(X,x)$ is abelian for $k\ge 2$.
Fact 2: ...
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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 ...
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Is the category of quotient of countably based topological spaces cartesian closed ?
In "Handbook of categorical algebra Vol 2" from Francis Borceux, the author gives a proof that $Top$ is not cartesian closed. It seems to me that this proof can be adapted to show that the category $\...
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Simultaneously minimizing intersections
This may be a standard problem in homotopy theory, but I don't know a good reference.
Let $\Sigma$ be a smooth, oriented surface, and let $X_1,X_2$ and $X_3$ be three smoothly embedded curves in $\...
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Has n^2*|sin(n)| limit? [closed]
Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity.
In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is ...
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Conditions under which a given scheme converges
I'm sorry in advance for how long this question is. Suppose I have a continuous function $f:\mathbb{R}^n \rightarrow \Delta_{n-1}$, where we think of the simplex $\Delta_{n-1}$ as the set
$\Delta_{n-...
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Continuity/measurability of a complicated extension of a family of continuous functions
Bonjour/bonsoir à tous et à toutes.
I've two questions related to something on which I'm working. I've already tried to discuss about them elsewhere, but it hasn't been fruitful so far.
Edit (4 Dic ...
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Why the category of core-compact spaces with continuous maps is not cartesian closed ?
According to ESCARDÓ-LAWSON-SIMPSON paper 'Comparing cartesian closed categories of (core) compactly generated spaces' The following four propositions are true:
A topological space $X$ is ...
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Local cartesian closedness in the category of compactly generated spaces
According the the nLab, the category of compactly generated (CG) spaces is not locally cartesian closed.
So if $A$ is a CG space and $C$ a CG space above $A$, $C$ may not be exponentiable.
What if we ...
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Whitney approximation without second countable
One version of Whitney's approximation theorem states the following:
Let $N$ be a smooth, Hausdorff, second-countable (or paracompact) manifold, then given any continuous function $F:N\to \mathbb{R}...
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Convergence in probability only depends on topology?
Suppose $(S,d)$ is a Polish space, and $X$, $(X_n)$ are random variables such that $X_n \to X$ in probability in $(S,d)$. Now suppose $d'$ is another metric on $S$, giving the same topology. Does $...
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Does X have any diagonal properties?
Assume that $2^{\omega_1}=2^\omega=\mathfrak{c}$. Let $D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$={$\xi<\mathfrak{c}:y(\xi)=1$}, the support of $y$, and let ...
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Properties of the Zariski-Riemann topology on the set of valuations
One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
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Topological proof of the Compactness Theorem in propositional logic without the Axiom of Choice
There is a well-known proof of the Compactness Theorem in propositional logic which uses the compactness of the space $\{0,1\}^P$, where $P$ is the set of propositional variables in consideration. In ...
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Higher dimensional Heegaard splittings?
Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing ...
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If a topological space X has $\aleph_1$-calibre, then it must be star countable?
If a topological space X has $\aleph_1$-calibre[definition], then it must be star countable?
What if the cardinality of the topological space X is additionally < = $2^{\aleph_0}$?
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Very General Topology
Suppose you take mathematical structures which have axioms based on sets and their subsets and you replace this with objects and subobjects, for example:
Let a very general topological space T be an ...
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on $F_\sigma$-discrete space
A space is $F_\sigma$-discrete space if it is the countable union of closed discrete subspaces. Is it true that every subset of an $F_\sigma$-discrete space is of the type $G_\delta$?
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A question about connectedness in Euclidean space [closed]
Here is a question which seems true to me but I can't rigorously show. Suppose $K$ is a compact subset of $\mathbb{R}^n$ such that $\mathbb{R}^n\setminus K$ is connected, does it follow that for any ...
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Name for a topological space where every closed set contains a closed point
A coauthor and I have stumbled upon a useful topological property -- namely, we are interested in the property that every nonempty closed set contains a closed point. However, neither of us are ...
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A question about connected inner limiting sets
Let M be a finite-dimensional Euclidean space or an infinite-dimensional separable Banach space.
An inner limiting subset of M is a countable intersection of open subsets of M-these sets are
usually ...
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Action on a compact group
If $G$ is an infinite compact group, how many orbits can $G$ have under the group action of its continuous automorphisms ?
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Question about 0-dimensional Polish spaces
Hello everybody,
I'm stuck with proving (or disproving) the following statement.
Statement:
For every $0$-dimensional Polish space $(X,\mathcal{T}\ )$, and a countable basis of clopen sets $\mathcal{...
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Special functions on the unit disk
Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.
We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a ...
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Sober except not $T_0$?
tl;dr: Is there an accepted or proposed term for a topological space whose $T_0$ quotient is sober?
The condition that a topological space be sober (and therefore equivalent to a locale) may be ...
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$G_\delta$-diagonal
Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't
a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence
${G_n}$ of ...
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Continuously selecting elements from unordered pairs
The symmetric square of a topological space $X$ is obtained from the usual square $X^2$ by identifying pairs of symmetric points $(x_1,x_2)$ and $(x_2,x_1)$. Thus, elements of the symmetric square can ...
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zeroset-diagonal
Is it true that a topology space X with a zeroset diagonal is first countable?
what if X is additionally CCC?
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On generalized ordered spaces
Let X be a Go space. If G is open in X, why is every convex component of G open?
( It is well known that any non-void subset G of X can be uniquely represented as a union
of its maximal convex ...
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locally connected versus locally compact
In the definition of a locally connected space we demand every neighbourhood of a point to satisfy certain condition whereas for a locally compact space we demand that one neighbourhood be there with ...
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When is a bijective map between bundles a homeomorphism?
Let $F \rightarrow E_i \rightarrow X_i$ be a bundle with fibre $F$ for i=1,2.
Let $f:E_1 \rightarrow E_2$ be a bijective continuous map and $h: X_1 \rightarrow X_2$ a homeomorphism.
Is f then also ...
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Topology generated by the collection of open sets
Hello, there is a statement as following:
If every point of X is a G_delta and X is T_1, then take Y = set of X,
plus the topology generated by all open sets needed to prove G_delta-ness of every ...
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