**33**

votes

**0**answers

4k views

### Grothendieck's manuscript on topology

Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis
Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are ...

**20**

votes

**0**answers

411 views

### Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...

**20**

votes

**0**answers

977 views

### Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property

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 ...

**19**

votes

**0**answers

487 views

### 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 ...

**18**

votes

**0**answers

587 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 ...

**17**

votes

**0**answers

490 views

### Can closed compacts in a topological group behave “paradoxically” with respect to unions, intersections, and one-sided translations?

Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$:
$A' = aA$,
$B' = bB$.
Suppose it is known that ...

**16**

votes

**0**answers

508 views

### Is there a category of topological spaces such that open surjections admit local sections?

The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of ...

**15**

votes

**0**answers

158 views

### Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer.
Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...

**15**

votes

**0**answers

505 views

### 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 ...

**15**

votes

**0**answers

1k views

### 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 $U^0,...,U^c$ ...

**14**

votes

**0**answers

549 views

### Which complete Boolean algebras arise as the algebras of projections of commutative von Neumann algebras?

Projections in an arbitrary commutative von Neumann algebra form a complete Boolean algebra.
Moreover, a morphism of commutative von Neumann algebras induces
a continuous morphism of the corresponding ...

**13**

votes

**0**answers

336 views

### Connected and locally connected, but not path-connected

Allow me to use some non-standard terminology:
A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to ...

**11**

votes

**0**answers

451 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 ...

**11**

votes

**0**answers

966 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 ...

**10**

votes

**0**answers

181 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 ...

**10**

votes

**0**answers

261 views

### When is the one-point compactification well-pointed?

This is a follow up to my previous
question.
Question:
Is there a reasonably natural set of conditions which guarantee that the one-point
compactification $X^+$ of a locally compact Hausdorff ...

**10**

votes

**0**answers

307 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 $S^2 \times M_2$?

**10**

votes

**0**answers

493 views

### Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every sigma-algebra the Borel algebra of a topology?
inspires the present question which asks for less.
Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ...

**10**

votes

**0**answers

526 views

### 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 ...

**10**

votes

**0**answers

599 views

### Characterization of Unusual Topologies of $\mathbb R$

Following some argument over a question on math.SE, I began to wonder:
We all know that in the standard topology there are no continuous bijections from $[0,1]$ to $(0,1)$ (for example by arguments ...

**9**

votes

**0**answers

112 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 ...

**9**

votes

**0**answers

425 views

### A group allowing exactly 7 group topologies

Is there a group $G$ allowing exactly 7 group topologies on $G$: $\mathcal T_{trivial}, \mathcal T_{discrete}, \mathcal T_1, \mathcal T_2,\mathcal T_3,\mathcal T_4, \mathcal T_5$ with
$$\mathcal ...

**9**

votes

**0**answers

285 views

### Topology of marked groups for different number of generators

A $k$-marked groups is a pair $(G,S)$ where $G$ is a group and $S$ is an ordered set of $k$ generators of $G$. Each such pair can be identified with a normal subgroup of the free group $F_k$ of rank ...

**9**

votes

**0**answers

283 views

### Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality

Is there a linearly Lindelöf space $X$ with
$\operatorname{cf} (L(X))> \aleph_{0}$
(where $L$ is the cardinal function Lindelöf degree)?
$L(X)$ must be a limit cardinal, like $\aleph_{\omega_{1}}$ ...

**9**

votes

**0**answers

489 views

### A proof of the gluing axiom of a TQFT

I posted the following question on math stackexchange but I have not received any answer.
So I hope people here can help me.
In the book Lectures on tensor categories and modular functors by Bakalov ...

**9**

votes

**0**answers

177 views

### H-spaces without rational homology

Does there exist a simply connected, non-contractible manifold $M$, which is an $H$-space,
and whose rational homology groups vanish in positive degrees?
My space $M$ is in fact homotopy equivalent ...

**9**

votes

**0**answers

575 views

### Does Urysohn's Lemma imply Dependent Choice?

It's widely known$^{1}$ that in the proof of Urysohn's Lemma (UL) one uses the Principle of Dependent Choice (DC). Inspired by the equivalence between DC and Baire's Category Theorem$^{2}$, I'd like ...

**8**

votes

**0**answers

80 views

### A 2 dimensional Sharkovskii type Theorem

Does there exist a homeomorphism of $\mathbb{R}^2$ with a periodic point of period three and no fixed points? Note that according to a theorem from Brouwer such homeomorphism must be orientation ...

**8**

votes

**0**answers

106 views

### The space of all compact metric spaces with Gromov-Hausdorff distance

Given two metric spaces $(X_1,d_1),(X_2,d_2)$ one can define $d_{GH}(X_1,X_2)$---the Gromov Hausdorff distance between them. It appears to be $0$ iff $X_1$ and $X_2$ are isometric. One can therefore ...

**8**

votes

**0**answers

170 views

### Is it possible to prove in ZF that a non-trivial compact connected Hausdorff space is uncountable?

Let $X$ be a compact, connected Hausdorff space with at least two points.
In $\mathrm{ZF}+\mathrm{AC}_\omega(\mathbb R)$, any countable compact Hausdorff space is metrizable, and from this it can be ...

**8**

votes

**0**answers

224 views

### Which topological spaces are coset spaces of locally compact groups?

What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ?
Such a space $X=G/H$ necessarily ...

**8**

votes

**0**answers

292 views

### Is the product of a discretely Lindelöf space with [0,1] discretely Lindelöf ?

A space $X$ is discretely Lindelöf iff given any discrete subset $D$ of $X$, its closure in $X$ is Lindelöf. Such spaces were introduced by Arkhangel'skii about 15 years ago (if I am not mistaken) ...

**8**

votes

**0**answers

526 views

### A basic question on Stone-Cech compactification of $\mathbb{Z}$

Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? ...

**8**

votes

**0**answers

532 views

### In ZF, when is a disjoint union of metrizable spaces metrizable?

It is easy to see that the disjoint union $\bigsqcup_i X_i$ of a collection of
metric spaces is metrizable, simply by rescaling or chopping off
the individual metrics to have diameter at most one, and ...

**8**

votes

**0**answers

486 views

### 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 ...

**8**

votes

**0**answers

685 views

### Is there a generalization of Brouwer's fixed point theorem?

In essence, this is the same problem as in
“The generalization of Brouwer's fixed point theorem?”.
But now I am determined to be careful. The main question is
the following:
Is there any ...

**8**

votes

**0**answers

354 views

### 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 ...

**7**

votes

**0**answers

253 views

### Two questions about universally measurable sets

I have two questions about universally measurable sets:
(1) Is there a universally measurable set of reals which does not have the Baire property?
(2) Is there a universally measurable set of reals ...

**7**

votes

**0**answers

131 views

### A forked plane continuum

I came up with this question while trying to solve the following MO one:
Does every connected set that is not a line segment cross some dyadic square?
Suppose $C$ is a plane continuum (i.e. a ...

**7**

votes

**0**answers

151 views

### How much of general topology can be developed by taking the notion of “connected set” as the sole topological primitive

Let X be an infinite regular topological space which is connected and locally connected. If no point of X is a cut point, does X always have base of connected open sets whose complements (with respect ...

**7**

votes

**0**answers

187 views

### Krull dimension and Morley rank

Definition : A Topological space $\mathcal{D}$ is called noetherian if it satisfies the descending chain condition for closed subsets. We define the dimension of $\mathcal{D}$ to be the supremum of ...

**7**

votes

**0**answers

283 views

### Is there in ZFC a topological space which is normal, ccc, countably compact, first countable and non-compact?

I am looking for a space as in the title and since many very similar spaces do exist in the literature, I wonder whether someone has a reference (different from the ones I cite below) or just some ...

**7**

votes

**0**answers

215 views

### Construct a topologically $\infty$-dimensional separable metric space.

But don't assume knowledge of any topological dimension theory. Here is a specific approach (an open problem):
Does there exist a separable metric space $X$ such that the following two conditions ...

**7**

votes

**0**answers

146 views

### Generalized Skorokhod spaces

Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...

**7**

votes

**0**answers

413 views

### Is the set of real-valued lower semi-continuous functions measurable in epigraph topology (= topology of Gamma convergence)?

Let LSC = LSC([0,1]) be the set of non-negative, lower semi-continuous functions on the unit interval which take values in $\mathbb{R}_+ \cup \{\infty\}$. We use epigraph topology on LSC, i.e. a ...

**7**

votes

**0**answers

313 views

### “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 ...

**7**

votes

**0**answers

250 views

### The self-duality of topological compactness

The impatient reader can skip my attempt at motivation and go straight my "Question formulations for the impatient."
In a failed(?) attempt at discovering something new, some years ago I toyed ...

**7**

votes

**0**answers

1k views

### Weak lower semi-continuity

Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will ...

**6**

votes

**0**answers

100 views

### Is $(\omega+1)^\omega$ with the box topology ultraparacompact?

Let $\omega+1$ be endowed with the interval topology, that is $U\subseteq (\omega+1)$ is open if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite. We call $U\subseteq (\omega+1)$ basic if ...

**6**

votes

**0**answers

160 views

### A compactification of the non-negative rationals with the discrete topology

Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is ...