Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ ...
Joseph O'Rourke's user avatar
24 votes
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907 views

The topologies for which a presheaf is a sheaf?

Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal. Suppose that $Q$ is a presheaf on $...
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Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property, without Choice

While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
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Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces?

Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \...
Vidit Nanda's user avatar
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Are amenable groups topologizable?

I've learned about the notion of topologizability from "On topologizable and non-topologizable groups" by Klyachko, Olshanskii and Osin (http://arxiv.org/abs/1210.7895) - a discrete group $G$ is ...
Łukasz Grabowski's user avatar
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554 views

What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?

Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
Noah Schweber's user avatar
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What is the Cantor-Bendixson rank of the space of first order theories?

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its ...
Danielle Ulrich's user avatar
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The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
Boaz Tsaban's user avatar
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Does there exist a continuous open map from the closed annulus to the closed disk?

(Originally from MSE, but crossposted here upon suggestion from the comments) In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
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What is the strongest nerve lemma?

The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology: If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...
2xThink's user avatar
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Čech functions and the axiom of choice

A Čech closure function on $\omega$ is a function $\varphi:\mathcal P(\omega)\to\mathcal P(\omega)$ such that (i) $X\subseteq\varphi(X)$ for all $X\subseteq\omega$, (ii) $\varphi(\emptyset)=\emptyset$,...
bof's user avatar
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Characterization of Fréchet-Urysohn spaces using sequential continuity at a point

A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$. $$x_n\to a \qquad \Rightarrow \qquad f(...
Martin Sleziak's user avatar
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"Next steps" after TQFT?

(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.) Recently, I've been ...
Nicholas James's user avatar
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Grothendieck dessins d'enfants - current surveys or text you can recommend?

I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph ...
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Is homeomorphism of simplicial complexes semidecidable?

Conventions: $\cong$ is homeomorphism of topological spaces and isomorphism of groups, $\equiv_G$ is the equality of two words over the generators of the group $G$. Simplicial complexes are finite. ...
Ville Salo's user avatar
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Spaces locally modelled on $L^2(\mathbb R)$

In this recent question, I learned that any two separable Banach spaces are homeomorphic. Based on some readings, I'm guessing that $L^2(\mathbb R)$ is homeomorphic to $\prod_{n=1}^{\infty} (0,1)$ (...
André Henriques's user avatar
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Is this "Homology" useful to study?

In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$. Now we can ...
Ali Taghavi's user avatar
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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 $U^0,...,U^c$ ...
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Which functions have all the common $\forall\exists$-properties of continuous functions?

This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well. For a ...
Noah Schweber's user avatar
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Do connected algebraic stacks have a smooth cover by a connected scheme?

An algebraic stack $X$ has an induced topological space $|X|$ given by equivalence classes of fields mapping to $X$ as outlined in the stacks project. If $|X|$ is connected, does that imply there ...
Leo Herr's user avatar
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Cardinality of the set of continuous functions

Suppose $(X,\tau)$ and $(Y,\sigma)$ are topological spaces. Let $F(X,Y)$ be the set of continuous functions $X\rightarrow Y$. I want to compute the cardinality of $F(X,Y)$. It depends not only on ...
Bugs Bunny's user avatar
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On projectively countable sets in the Hilbert cube

A subset $A$ of a topological space $X$ is called projectively countable if for any continuous map $f:X\to\mathbb R$ the image $f(A)$ is countable. It is easy to see that each projectively countable ...
Taras Banakh's user avatar
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Small cardinals related to topological convergence

Recall that a topological space is called sequential if a set is closed if and only if it contains all limits of convergent sequences lying inside of it. A space $X$ is called Frechet if for every non-...
Santi Spadaro's user avatar
14 votes
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401 views

Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?

Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
Taras Banakh's user avatar
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A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
Taras Banakh's user avatar
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13 votes
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635 views

Covering number estimates for Hölder balls

Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
ABIM's user avatar
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Covering image of a connected CW-complex need not be a CW-complex

This question is already asked here MSE, and there is an answer based on some conjecture (probably still open). I am posting the same question for a counterexample (if any, not based on such unsolved ...
Sumanta's user avatar
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252 views

Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane. The following questions are motivated by Anton Petrunin's Disc bounded ...
Wlodek Kuperberg's user avatar
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512 views

pizza lemma (topology)

given six real-analytic arcs in the unit disk $D$, each of which connects the origin to a boundary point, and no two arcs meet anywhere except at the origin, and the arcs meet at equal (60 degree) ...
Michael Beeson's user avatar
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273 views

Why must commuting maps (of an interval) without common fixed points have at least 11 fixed points for the composition?

I've been looking at the examples of commuting functions on a closed interval which have no common fixed points. These were discovered in 1967 by William M Boyce and J Philip Huneke. Earlier work by ...
Jeff Norden's user avatar
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561 views

Multiplicity of ball covering

Background. My questions are motivated by the following: A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete and Computational Geometry, 8 (1992) 109-130) considered the ...
Misha's user avatar
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12 votes
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Do compact inverse-property loops (or just compact Moufang loops) have bi-invariant Haar measure?

So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...
Harry Altman's user avatar
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Metric completion of an algebraically closed field is algebraically closed?

Let $F$ be a complete metric topological field. Suppose there is a subfield $F_1 \subset F$, algebraically closed and topoolgically dense in $F$. Must $F$ itself be algebraically closed? We can ...
Gerald Edgar's user avatar
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12 votes
0 answers
239 views

Is there a characterization of the class of first-order formulas that are closed in every compact Hausdorff structure?

Fix a relational language $\mathcal{L}$. (I don't think relational really matters that much but I don't want to worry about it.) A topological $\mathcal{L}$-structure is an $\mathcal{L}$-structure $M$ ...
James Hanson's user avatar
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12 votes
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286 views

Topology is to semi-decidability, coarse structures are to what?

There is a folklore correspondence between topology as semi-decidability amongst computer scientists, which is explained in places like: The monograph Synthetic Topology: of Data Types and Classical ...
Siddharth Bhat's user avatar
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249 views

Counter example to lifting contractibility of a topological space

I'm looking for a simple example of an open proper continuous map between topological spaces $\varphi:X\to Y$ such that : $Y$ is contractible and locally contractible ; for any $y\in Y$, $\varphi^{-1}...
thibaud lemanissier's user avatar
12 votes
0 answers
171 views

A connected Borel subgroup of the plane

It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...
Taras Banakh's user avatar
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12 votes
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367 views

Does each compact topological group admit a discontinuous homomorphism to a Polish group?

A compact topological group $G$ is called Van der Waerden if each homomorphism $h:G\to K$ to a compact topological group is continuous. By a classical result of Van der Waerden (1933) the groups $SO(...
Taras Banakh's user avatar
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12 votes
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For a Banach space $X$, when is $X$ homeomorphic to $X \setminus A$?

$\mathbb{R}^n\not\cong\mathbb{R}^n\setminus\{0\}$ are not homeomorphic is a triviality from Algebraic Topology. On the other hand, if $X$ is an infinite dimensional Banach space, then $X \cong X\...
T. Amdeberhan's user avatar
12 votes
0 answers
379 views

L-spaces without convergent sequences

An L-space is a regular hereditarily Lindelof space which is not hereditarily separable. Consistent examples of L-spaces are relatively easy to come by (for example, Suslin Lines), but the first ...
Santi Spadaro's user avatar
12 votes
0 answers
215 views

Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?

Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table. Let $$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$ for all $n\in\mathbb{N}$. Then since $C_{n}$ is a ...
Joseph Van Name's user avatar
12 votes
0 answers
1k views

Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
Lasse Rempe's user avatar
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12 votes
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392 views

Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality

A space is linearly Lindelöf iff every open cover $C$ has a subcover $S$ with $\operatorname{cf} (|S|)= \aleph_{0}$. Question. Is there a linearly Lindelöf space $X$ with $\operatorname{cf} (L(X))...
Alberto Levi's user avatar
12 votes
0 answers
457 views

3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?
Murat Saglam's user avatar
12 votes
0 answers
1k views

Paracompact Hausdorff but not compactly generated?

I'm sorry to be asking a (possibly) elementary question, but I've run into a problem in point-set topology; I've just read that there exists paracompact Hausdoff spaces which are not compactly ...
David Carchedi's user avatar
11 votes
0 answers
284 views

What is known about differentiable and analytic structures on the long line (and half-line)?

When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ ...
Gro-Tsen's user avatar
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11 votes
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514 views

Is every Baire metric space a complete metric space in disguise?

I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here: Let's say that a metric space $X$ is Baire if every countable ...
fedja's user avatar
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11 votes
0 answers
318 views

Does any real function have a Lipschitzian restriction on $D$?

Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
Dattier's user avatar
  • 3,737
11 votes
0 answers
153 views

Embedding of regular spaces in countably compact Hausdorff spaces

It is well-known that each completely regular space embeds into a compact Hausdorff space. Problem. Is it true that each regular space embeds into a countably compact Hausdorff space? The problem ...
Lviv Scottish Book's user avatar
11 votes
0 answers
324 views

If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?

It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle? ...
James Baxter's user avatar
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