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34 votes
4 answers
9k views

Why are the integers with the cofinite topology not path-connected?

An apparently elementary question that bugs me for quite some time: (1) Why are the integers with the cofinite topology not path-connected? Recall that the open sets in the cofinite topology on a ...
Theo Buehler's user avatar
  • 5,743
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 ...
Justin Campbell's user avatar
12 votes
4 answers
1k views

What was Burroni's sketch for topological spaces?

In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...
Kevin Carlson's user avatar
9 votes
4 answers
1k views

When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)

The title has it all. I'm looking for a reference to the following: Q. Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the ...
Salvo Tringali's user avatar
24 votes
2 answers
4k views

complement of a totally disconnected closed set in the plane

While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
Folkmar Bornemann's user avatar
16 votes
2 answers
820 views

Klee's trick --- more applications

In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
Anton Petrunin's user avatar
16 votes
2 answers
4k views

Is there a "disjoint union" sigma algebra?

I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally: For an indexed family of sets $\{A_i\...
Neil Toronto's user avatar
7 votes
2 answers
608 views

What is the name for a set endowed with a Lipschitz structure?

I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...
Taras Banakh's user avatar
  • 41.9k
7 votes
1 answer
183 views

Stability Question for Isotopies Between Compact Sets

Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$. Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
John Samples's user avatar
3 votes
0 answers
78 views

Nowhere dense covering number of a connected $T_2$ space

This is a generalization of an older question. If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a ...
Dominic van der Zypen's user avatar
3 votes
4 answers
934 views

Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1. Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $...
Mirko's user avatar
  • 1,375
2 votes
0 answers
159 views

Are there hereditarily square-boxed plane continua?

A plane continuum is a bounded, closed and connected subset of the plane. A bounding box $B$ for a plane continuum $C$ is a rectangle $B=[a,b]\times[c,d]$ (including sides and interior) such that $C$ ...
Mirko's user avatar
  • 1,375
1 vote
1 answer
388 views

About isotopy of simple close curve

In the Primer mapping class group by farb Margalit. We have : Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if ...
T566y65tt's user avatar
  • 119
29 votes
2 answers
2k views

Contractibility of the space of Jordan curves

Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$. If the curves are ...
Mohammad Ghomi's user avatar
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 ...
Bruno Stonek's user avatar
  • 3,004
23 votes
3 answers
2k views

An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request

There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter. But there's ...
Tom Leinster's user avatar
  • 27.7k
21 votes
7 answers
1k views

Reference for topological graph theory (research / problem-oriented)

I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
21 votes
1 answer
2k views

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
17 votes
1 answer
989 views

Can two-point sets be Borel?

Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914. I wonder if the following question of ...
Mohammad Golshani's user avatar
17 votes
2 answers
2k views

Foundations of topology

I recently went to a talk of Oleg Viro where he expressed his dissatisfaction with current foundations of differential topology parallel to what has been discussed here. Also some time ago I read ...
Bananeen's user avatar
  • 1,190
17 votes
10 answers
3k views

References for homotopy colimit

(1) What are some good references for homotopy colimits? (2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...
Kevin Walker's user avatar
  • 12.8k
16 votes
5 answers
3k views

Compactness of the Hilbert cube without the Axiom of Choice

I am just curious: is there a published proof of the compactness of the Hilbert cube that does not use the Axiom of Choice, or is it well known?
Alexey Muranov's user avatar
14 votes
2 answers
761 views

Is there a large colimit-sketch for topological spaces?

Question. Is there a large colimit-sketch $\mathcal{S}$ such that $\mathrm{Mod}(\mathcal{S}) \simeq \mathbf{Top}$? In other words, is there a category $\mathcal{E}$ with a class of cocones $\mathcal{S}...
Martin Brandenburg's user avatar
14 votes
5 answers
2k views

Largest Hausdorff quotient

The inclusion of the full subcategory of Hausdorff topological spaces into the category of topological spaces has a left adjoint, which can be proven easily by the Adjoint Functor Theorem (see for ...
mbasic's user avatar
  • 143
14 votes
1 answer
937 views

Classification of 3-dimensional manifolds with boundary

It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as $$\mathcal{M}=P_{1}\#\dots\# P_{n}$$ where $P_{i}$ are prime manifolds, i.e. ...
G. Blaickner's user avatar
  • 1,429
13 votes
1 answer
570 views

Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects

These are five important constructions and I would like to know how they are related. The $n$th unordered configuration space of a space $X$ is $$ \operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
მამუკა ჯიბლაძე's user avatar
13 votes
3 answers
357 views

How should one look at the set of compatible ring structures on a given group?

Earlier today I had a conversation with a friend about ways of putting topologies on sets of first-order structures; we wound up talking about reducts and expansions from a topological point of view, ...
Noah Schweber's user avatar
13 votes
1 answer
736 views

Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
Justin Moore's user avatar
  • 3,547
13 votes
5 answers
1k views

A generalization of metric spaces

Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i....
Monroe Eskew's user avatar
  • 18.6k
12 votes
4 answers
2k views

Early illustrations of topological notions in published work

Cross-posted from HSM: I posted this question a bit more than a week ago but have not gotten any answers at HSM. The only comment on the posting asks if I would accept polyhedral pictures ...
Sam Nead's user avatar
  • 28.2k
10 votes
2 answers
270 views

Which compact metrizable spaces have continuous choice functions for non-empty closed sets?

Let $X$ be a compact metrizable space and let $\mathcal{K}_{ne}(X)$ be the collection of non-empty closed subsets of $X$ with the Vietoris topology (i.e. the topology induced by the Hausdorff metric ...
James E Hanson's user avatar
10 votes
1 answer
695 views

Topology from the viewpoint of the filter endofunctor

Question. Are there any references that develop general topology from the viewpoint of a functor $$\Phi : \mathbf{Rel} \rightarrow \mathbf{Rel}$$ that assigns to every set $X$ the set $\Phi(X)$ of ...
goblin GONE's user avatar
  • 3,793
10 votes
0 answers
441 views

A new $\ell_p$-metric on the hyperspace of finite sets?

Let $(X,d)$ be a metric space and $Fin(X)$ be the family of all non-empty finite subsets of $X$. For every $n\in\mathbb N$ the elements of the power $X^n$ are thought as functions $f:n\to X$ where $n:=...
Taras Banakh's user avatar
  • 41.9k
10 votes
3 answers
2k views

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?
Louis A's user avatar
  • 360
10 votes
2 answers
750 views

Is there a compact space with no countably generated dense subspace?

This is a reformulation of this MO question which recieved little or no attention due to the fact that the OP gave no motivation whatsoever. I found the question quite interesting and decided to give ...
Ramiro de la Vega's user avatar
9 votes
1 answer
1k views

A question concerning separate and joint continuity of bilinear maps

Suppose that $V$ is a locally convex topological vector space and $f:V^2 \to V$ is a bilinear map. Suppose that $C \subseteq V$ is compact and convex, $f$ maps $C^2$ into $C$ and $f \restriction C^2$ ...
Justin Moore's user avatar
  • 3,547
9 votes
0 answers
211 views

Is the category of all topological spaces, including the bad ones, simplicially tensored and cotensored?

Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones. We can make $\textbf{Top}$ into a simplicially enriched category as follows: Given topological spaces $X$ and $Y$,...
Zhen Lin's user avatar
  • 15.9k
9 votes
2 answers
239 views

Hausdorff open image of a Polish space

Let $f\colon X\to Y$ a continuous open and surjective function, where $X$ is Polish. It is known that $Y$ is Polish if: $f$ is closed or $Y$ is metric. Suppose that we know that $Y$ is Hausdorff, ...
Hector Pinedo's user avatar
9 votes
1 answer
322 views

What is the (genuine) name for the Gutik hedgehog?

Working with non-regular topological semigroups, my collegue Oleg Gutik discovered a special space $H$ which we named Gutik's hedgehog. It is homeomorphic to the space $$H:=\{(0,0)\}\cup\{(\tfrac1n,0):...
Taras Banakh's user avatar
  • 41.9k
9 votes
0 answers
569 views

A standard name for a function satisfying the intermediate value theorem?

Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property: $(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
Taras Banakh's user avatar
  • 41.9k
9 votes
2 answers
772 views

Surreal compactness

In a comment here, Joel David Hamkins said: ...I think perhaps every set-sized open cover of a bounded interval in the surreals has a finite subcover, but there are proper class open covers with no ...
Gerald Edgar's user avatar
  • 41.1k
8 votes
2 answers
2k views

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 ...
Guillaume Brunerie's user avatar
8 votes
4 answers
6k views

Connectedness and the real line

It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be ...
Daniel Barter's user avatar
7 votes
2 answers
722 views

Consistency of a strange (choice-wise) set of reals

Consider a set $X\subseteq \mathbb{R}$ such that $X$ is not separable wrt its subspace topology For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$ In a ...
Lorenzo's user avatar
  • 2,286
7 votes
1 answer
181 views

Lachlan on topology for priority arguments

There is a set of notes by Lachlan from 1973 on casting priority arguments in topological language; references to these notes are few and far between, but one source refers to them as "Topology for ...
Noah Schweber's user avatar
7 votes
1 answer
399 views

Objects whose morphisms are Lipschitz maps

I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz"). The answer seems pretty clear, and proceeds like the definition of manifolds: 1) If $...
Benoit Jubin's user avatar
6 votes
1 answer
339 views

Factorization of a certain map through a CW-complex

Suppose that $X$ is a paracompact Hausdorff space (e.g. a metric space) with $\dim X=n$ (the Lebesgue covering dimension). I want to find a proof (or a reference) that any (continuous) map $f: X \to K(...
Ilja's user avatar
  • 423
6 votes
0 answers
309 views

Have we discovered constructions for natural fractional dimensional spheres?

I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
Sidharth Ghoshal's user avatar
6 votes
1 answer
478 views

Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets?

Is the following statement true, and if it is, does someone have a reference? Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, =...
Gro-Tsen's user avatar
  • 32.5k
5 votes
1 answer
206 views

If a subspace $F$ is contained in a subspace $G$, and $H$ is close to $G$, can we choose a subspace of $H$ close to $F$?

Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a gap: if $G$ ...
erz's user avatar
  • 5,529