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Questions tagged [gn.general-topology]

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

4
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1answer
338 views

When is the generalized Cantor space $\kappa$-compact?

My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references. The generalized Cantor space is the space $2^\kappa$, with basic open ...
4
votes
2answers
617 views

How to define compatible topology for first-order structures?

Background Because a bounded distributive lattice can be represented by the clopen sets of a Priestley space, I tried to learn some basics about Priestley spaces. After reading (on Wikipedia) A ...
12
votes
1answer
706 views

Ultralimit versus partial limit

Let $\omega$ be a nonprincipal ultrafilter on $\mathbb N$. A standard construction gives an $\omega$-limit, say $x_\omega$, for any bounded sequence $(x_n)$ of real numbers. Namely, there is unique ...
8
votes
0answers
245 views

Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?

There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$. We usually call it $\mathbb{C}$, but by this we impose a ...
8
votes
2answers
807 views

Is Stone-Čech compactification of 0-dimensional space also 0-dimensional?

What is an example of a 0-dimensional locally compact Hausdorff space $X$ for which the Stone-Čech compactification $\beta(X)$ is not 0-dimensional? It is known that if $X$ is a 0-dimensional locally ...
5
votes
1answer
219 views

Is each compactification of $\mathbb N$ soft?

Definition. A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be soft if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\...
5
votes
2answers
228 views

Is each locally compact group topology on the permutation group discrete?

Question. Is each locally compact group topology on the permutation group $S_\omega$ discrete? Here $S_\omega$ is the group of all bijections of the countable ordinal $\omega$. A group topology on a ...
3
votes
1answer
133 views

Maximal elements in the partially ordered set of image spaces

If $(X,\tau)$ is a topological space, let $\text{Im}(X)$ denote the collection of subsets $S$ of $X$ such that there is a continuous function $f:X\to X$ with $\text{im}(f) = S$. Is there a space $(X,\...
3
votes
1answer
139 views

Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods

Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?
3
votes
1answer
596 views

Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\...
2
votes
1answer
148 views

Are the connected components of a Priestley space closed?

Preliminaries A Priestley space is both a poset and a topological space. The topologically connected components of the space are trivially closed. (They are just the points of the underlying set.) But ...
2
votes
0answers
72 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 ...
2
votes
1answer
164 views

Maximal connected topologies

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected. If $(X,\tau)$ is ...
1
vote
1answer
113 views

Two consecutive continua

Are there two non homeomorphic continua $X,Y$ such that $X $ can be embedded in $Y$ but there is no topological space $Z$ with $$X<Z<Y.$$ The later relation means that $Z$ ...
1
vote
1answer
147 views

Totally non fixed point property

Edit: According to the comment of Pietro Majer, I revise the question Is there a non singleton compact connected Hausdorff topological space $X$ for which the following property hold?: "Constant ...
1
vote
1answer
115 views

Is every semi-stratifiable space $\omega$-monolithic?

Is every semi-stratifiable space $\omega$-monolithic? Definitions A topological space $(X,\tau)$ is called semi-stratifiable if there exists a function $g:\omega\times X\to\tau$ such that: for any ...
1
vote
1answer
136 views

Two questions about the extent to which simple arcs and simple closed curves can fill up higher dimensional Euclidean spaces

For each positive integer n, let E(n) be n-dimensional Euclidean space with its standard metric and let p(n) be some fixed point of E(n). The so-called "Osgood Curve" shows that there can exist simple ...
0
votes
1answer
132 views

does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and ..?

Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with no-zero finite length $L$. Then, does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
-10
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1answer
1k views

Union of uniformly connected sets

I will call a set uniformly connected regarding some uniform space when it is connected regarding every entourage of this uniform space (entourages are considered as digraphs and it is taken strong ...
180
votes
3answers
10k views

Is $\mathbb R^3$ the square of some topological space?

The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...
104
votes
3answers
7k views

Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
45
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5answers
4k views

Does homology have a coproduct?

Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
35
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8answers
3k views

What is a metric space?

According to categorical lore, objects in a category are just a way of separating morphisms. The objects themselves are considered slightly disparagingly. In particular, if I can't distinguish ...
41
votes
7answers
11k views

Is there a measure zero set which isn't meagre?

A subset of ℝ is meagre if it is a countable union of nowhere dense subsets (a set is nowhere dense if every open interval contains an open subinterval that misses the set). Any countable set ...
32
votes
14answers
4k views

What are interesting families of subsets of a given set?

Motivation The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$. Indeed, one defines a topology on $S$ to be a family of subsets ...
30
votes
14answers
11k views

What is your favorite proof of Tychonoff's Theorem?

Here is mine. It's taken from page 11 of "An Introduction To Abstract Harmonic Analysis", 1953, by Loomis: http://www.archive.org/details/introductiontoab031610mbp http://ia331316.us.archive.org/3/...
53
votes
3answers
5k 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? UPDATE1. It is a duplicate of the question here https://math.stackexchange.com/...
34
votes
3answers
4k views

Why do finite homotopy groups imply finite homology groups?

Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\...
18
votes
6answers
8k views

Is a topology determined by its convergent sequences?

Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the ...
32
votes
7answers
5k views

Is there a Whitney Embedding Theorem for non-smooth manifolds?

For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? (i.e. Can ...
15
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3answers
2k views

Physical interpretations/meanings of the notion of a sheaf?

I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
34
votes
1answer
2k 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 ...
25
votes
2answers
2k views

Does Arzelà-Ascoli require choice?

Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version: ...
23
votes
12answers
4k views

Classic applications of Baire category theorem

I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
23
votes
6answers
3k views

Least number of charts to describe a given manifold

Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it. E.g. a circle requires at least two charts, and ...
18
votes
4answers
5k views

Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?

An anonymous question from the 20-questions seminar: Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?
22
votes
6answers
2k views

Is there a topological description of combinatorial Euler characteristic?

There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
15
votes
9answers
2k 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 ...
24
votes
3answers
4k views

Is “compact implies sequentially compact” consistent with ZF?

Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess ...
36
votes
1answer
1k views

Are there only countably many compact topological manifolds?

Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...
23
votes
1answer
1k views

Is there a topological space X homeomorphic to the space of continuous functions from X to [0, 1]?

In general, we might ask when we can find interesting spaces $X, Y$ such that $X$ is homeomorphic to $[X, Y]$. By the Lawvere fixed point theorem $Y$ must have the fixed point property. Happily, $Y = [...
23
votes
1answer
558 views

Which ordered fields are homeomorphic to their power?

It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ...
16
votes
3answers
2k views

Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently: In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations $\...
36
votes
1answer
2k views

Is every connected scheme path connected?

Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following. Let's ...
21
votes
2answers
1k views

An order type $\tau$ equal to its power $\tau^n, n>2$

(This is a re-post of my old unanswered question from Math.SE) For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types. Recall that: $...
19
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1answer
989 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 ...
19
votes
2answers
1k 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 ...
18
votes
4answers
2k views

When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
17
votes
4answers
2k views

What is the “right” universal property of the completion of a metric space?

I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes: Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...
16
votes
4answers
4k views

Unique limits of sequences plus what implies Hausdorff?

It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff. What I am ...