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

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2
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196 views

Sequences and pseudocharacter in compact spaces

Is there a consistent example of a compact Hausdorff space $X$ on which the following holds? i) there is a $Y \in {[X]}^{\aleph_1}$ such that $\psi (Y) = \aleph_1$; and ii) there is no non-trivial ...
35
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3answers
1k views

If any open set is a countable union of balls, does it imply separability?

If a metric space is separable, then any open set is a countable union of balls. Is the converse statement true?
3
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0answers
37 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 ...
23
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1answer
720 views

Computing Self-Intersections with Complex Analysis

It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis: $$n = \oint_C\frac{dz}{z}.$$ You can also count the number of roots of $f(z) = 0$ inside a close ...
6
votes
1answer
184 views

Immersion of $S^1$ in $\mathbb{R}^2$ that can be extended to $\mathbb{D}$

I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow ...
3
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0answers
81 views

The Klee Trick for subsets of $\mathbb{R}^3$

I asked the question Is dimension given by the Klee trick ever sharp? That question remains unanswered, so I thought I might ask a slightly more concrete question along those lines. Given a metric ...
4
votes
1answer
149 views

Totally disconnected subspaces

This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then. Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it ...
4
votes
2answers
73 views

Measure on hyperspace of compact subsets

For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets ...
3
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1answer
100 views

Lipschitz function with somewhere dense image

Let $Q=[-1,1]^2$ denote the unit square and let $f:Q\to Q$ be a Lipschitz function such that for any ball $B(a,r)\subset Q$ with radius $r$, the width of the image $f(B(a,r))$ is at least $cr$ for ...
10
votes
1answer
582 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, ...
0
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1answer
51 views

Which Hyperspace Topologies Yield Topological Lattices?

At least on a continuum, the binary operations of intersection and union are Vietoris-continuous. But the Vietoris topology only applies the the collection of NONEMPTY closed subsets, and this means ...
22
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8answers
1k views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful? I have a vague idea of the possibility of ...
5
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0answers
159 views

Distributivity of group topologies on $\Bbb Z$

Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$. It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice. Is $(\mathcal L,\subseteq)$ distributive?
0
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1answer
66 views

Can we build a continuous function from “fibers”/preimages defined over a topological base?

I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...
16
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2answers
913 views

Colimits in the category of smooth manifolds

In the category of smooth real manifolds, do all small colimits exist? In other words, is this category small-cocomplete? I can see that computing push-outs in the category of topological spaces of ...
3
votes
1answer
228 views

Decomposition vs filtration vs stratification

Are there accepted/standard definitions of "decomposition", "filtration", and "stratification" of a topological space (or of a manifold, or of an algebraic variety) $X$? I tend to understand ...
2
votes
0answers
34 views

Local cross sections in infinite dimensional groups

Let $G$ be an (infinite dimensional) compact connected abelian group and $H$ be a closed subgroup of $G$. The quotient morphism $G\to G/H$ may not possess a local cross section, there are examples ...
0
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0answers
55 views

Branch point and alexandrov embeddedness

This is a question I have asked on mathstackexchange with a bounty but without any answer; it is probably more adapted to mathoverflow: Let us assume that $\Sigma_n$ is a sequence of topological ...
7
votes
1answer
400 views

Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{-1}$ continuous at $(1,1)$

Let $(G,\mathcal T)$ be a completely regular topological space. Is there a group structure on $G$ such that the function $$f:G\times G\to G$$ $$f(x,y)=xy^{-1}$$ is continuous at $(1,1)$?
5
votes
1answer
659 views

A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: $X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite ...
17
votes
2answers
656 views

Which are the rigid suborders of the real line?

Which are the rigid suborders of the real line? If A is any set of reals, then it can be viewed as an order structure itself under the induced order (A,<). The question is, when is this structure ...
6
votes
3answers
352 views

Binary relations as the topological closure of the diagonal

If $(X,\tau)$ is a topological space, we can consider the product topology on $X\times X$ and take the closure of the diagonal $\Delta_X = \{(x,x): x\in X\}$, which we denote by ...
24
votes
3answers
1k views

Proving that a function's image contains (1/n,…,1/n)

This question is a follow-up to a previous question answered by Neil Strickland: Map from simplex to itself that preserves sub-simplices Let $B$ denote the closed unit ball in $\mathbb{R}^2$ and let ...
14
votes
2answers
460 views

A “dimension” for Tychonoff spaces

It's well-known that any Tychonoff space $X$ can be embedded in $[0,1]^k$ for some cardinal $k$. It's natural to ask what the smallest such $k$ is (let's call it $k(X)$). However, this probably ...
9
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1answer
522 views

Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory? I refer to the (systematic, formalist) study of the category of sheaves on a site or the study of topology in a ...
0
votes
1answer
93 views

Dimension of Inverse image

Suppose we have a smooth map between two smooth manifolds $f:M→N$ such that $\dim M>\dim N$, and let $p∈N$ be a critical value. Suppose we know that $X:=f^{-1}(p)$ is a differentiable manifold (or ...
8
votes
2answers
314 views

Ultracoproducts and Cartesian products

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$. As a C$^*$-algebraist, I ...
0
votes
1answer
65 views

subspace in pseudotopological space

Every topological space gives rise to a pseudotopological space. Conversely, if $X$ is a pseudotopological space then we can define a topology on $X$ such that every filter converging to $x$ in the ...
6
votes
0answers
136 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 ...
3
votes
0answers
96 views

Decreasing sequence of closed convex sets in a Banach space

Let $(C_n)$ be a decreasing sequence of closed convex subsets in a Banach space $(E,\Vert \cdot \Vert)$. The question I have is about the content of $C=\bigcap_{n=0}^\infty C_n$. If the $C_n$ are ...
4
votes
1answer
127 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, ...
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0answers
27 views

The topology of complete minimal surfaces of finite total Gaussian curvature [closed]

Suppose that M is a complete minimal surface with finite total curvature. If M is embedded in $\mathbf{R}^3$, then we observe that M viewed from infinity looks like a plane passing through the ...
7
votes
1answer
265 views

Topological conditions forcing continuity

Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous. Question: Under what ...
10
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1answer
226 views

Maximum length of a chain of topologies on $\Bbb R$

Let $\frak T$ be a totally ordered set of topologies on $\Bbb R$. Is $|\frak T|\le |\Bbb R|$?
7
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4answers
621 views

Defining a topology in the Power Set

I have the follwing question: Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$. If the ...
0
votes
1answer
89 views

Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem. Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...
7
votes
3answers
653 views

Is the reals the smallest connected ordered topological ring?

The real numbers is a locally compact Tychonoff connected complete ordered topological field. I am looking at minimal collections of adjectives that can characterize the reals. The one often used to ...
6
votes
2answers
293 views

A continuous notion of realizability

I have been interested in non-classical logics, off and on, for quite a while. This question is probably very basic, and I hope it is not too low-level for MO. My question stems from an attempt to ...
3
votes
0answers
125 views

$S^{3}$-valued harmonic analysis

Edit: Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider $$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid ...
22
votes
1answer
1k views

Fake versus Exotic

Without recourse to the Disc Theorem (or its progeny), is it true that all known examples of exotic differentiable structures on 4-manifolds would be fake rather than exotic? Terminology (perhaps ...
4
votes
1answer
165 views

Are infinite groups “locally topologizable”?

Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point? The question is inspired by and related to ...
21
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2answers
865 views

A question about “Zariski dense” arguments

This question is a little basic, but I think it is consistent with the goals of MO. My question is about a certain type of argument in algebraic geometry which exploits the abundance of dense sets ...
2
votes
1answer
63 views

Separating Differences of Open Sets

Has anyone ever considered something like the following separation axiom? $(*)$ For any pair of open sets $O$ and $N$, there exist disjoint open sets containing $O\setminus N$ and $N\setminus O$. ...
19
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0answers
464 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 ...
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0answers
141 views

Continuous dependence of the roots of a polynomial on its coefficients

In their article "The roots of a polynomial vary continuously as a function of the coefficients" Gary Harris and Clyde Martin give a topological proof of the well-known theorem that the roots of a ...
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0answers
52 views

Ostaszewski space's construction Lemma

I'm studying the Ostaszewski's article "On Countably Compact, Perfectly Normal Spaces". I'll add some context. Lemma 1.2 says the following: Let $X$ be a locally compact, zero-dimensional and ...
7
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0answers
385 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 ...
6
votes
1answer
164 views

Finiteness as a motivation for compactness

Another history question, and I am not sure if I will get any answers. (If anyone knows of a good history of math list to use for this question I would be happy for any tips. The one I used to post to ...
8
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0answers
457 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 ...
2
votes
1answer
90 views

A Property of Baire Spaces

I posed this on August 8 on math.stackexchange, but there has been no response. Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is ...