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

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29
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3k views

Grothendieck's manuscript on topology

Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible ...
18
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0answers
537 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
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453 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 ...
15
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471 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 ...
14
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343 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 ...
14
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454 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 ...
13
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503 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 ...
12
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220 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
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928 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
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244 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
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331 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 ...
10
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570 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
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155 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 ...
9
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255 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
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248 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
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285 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 $S^2 \times M_2$?
9
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168 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
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507 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 ...
8
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285 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
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465 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 ...
8
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487 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
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495 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
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527 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
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288 views

Reference request for TQFT, functoriality

I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds. It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says; The function $(M, \partial_{-}M, ...
8
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447 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
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0answers
674 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
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349 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
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120 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
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151 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 ...
7
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180 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 ...
7
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246 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
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199 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
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127 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
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344 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
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229 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 ...
6
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0answers
72 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 ...
6
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137 views

Zariski-homeomorphisms

This question is motivated by two questions at MO and at MSE. I am interested in homeomorphism types of (irreducible) complex-projective varieties with respect to the Zariski topology. Any two ...
6
votes
0answers
258 views

Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure. Notation used ...
6
votes
0answers
88 views

How to call a point in a space having the property that there is essentially one $\omega$-sequence converging to it?

Consider the point $x=\langle \omega_1,\omega\rangle$ in the Tychonov plank $(\omega_1 + 1)\times(\omega + 1)$. Then there is essentially only one sequence (of length $\omega$) converging to it, ...
6
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223 views

Is there Ultracoproduct-like construction for topological spaces in general?

In http://arxiv.org/pdf/math/9704205.pdf they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
6
votes
0answers
194 views

Spaces that never separate the Hilbert cube

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement. Any finite dimensional space has this ...
6
votes
0answers
277 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 ...
6
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228 views

How to use the Lefschetz trace formula on infinite dimensional spaces?

I think the Lefschetz trace formula says something like: if $F: X \to X$ is a continuous map of compact manifolds, then $\chi(X^F) = \sum (-1)^i \mathrm{Tr} f_*|_{H_i(X)}$ First of all, this ...
6
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293 views

The Space of Cellular Maps

Let $X$ and $Y$ be CW complexes. Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...
6
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396 views

Ever seen a ringed group?

A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
6
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388 views

The Mapping Cylinder of a Pullback Square

Suppose I have a pullback square, which I think of as a map from the fibration $q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$ from the mapping cylinder $M$ of ...
6
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198 views

In a locally contractible space can we find local bases of contractible sets whose closures are locally contractible?

In a locally contractible topological space $X$ is it possible at each point $x$ to find a local basis of contractible sets $U_i\ni x$ such that the closure of each set $\overline{U_i} \subset X$ is ...
6
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0answers
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 ...
5
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183 views

Homeomorphisms of product spaces: an example

In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to ...
5
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83 views

Tensor product of dual groups

Let $G,H$ be compact abelian groups, $G^*,H^*$ be their Pontryagin duals, $G^*\otimes H^*$ the tensor product of $G^*,H^*$ and $K=(G^*\otimes H^*)^*$. Does the group $K$ have a special name? What is ...