# Questions tagged [gn.general-topology]

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

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### 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

**2**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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\...

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votes

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

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votes

**1**answer

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

**3**answers

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**

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**3**answers

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**

votes

**5**answers

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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**8**answers

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

**7**answers

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**

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**14**answers

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

**14**answers

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

**3**answers

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

**3**answers

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. $\...

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votes

**6**answers

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**

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**7**answers

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**

votes

**3**answers

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**

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**1**answer

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**

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**2**answers

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**

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**12**answers

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**

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**6**answers

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 ...

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**4**answers

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**

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**6**answers

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 ...

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votes

**9**answers

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

**3**answers

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

**1**answer

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

**1**answer

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 = [...

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votes

**1**answer

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 ...

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**3**answers

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

**1**answer

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 ...

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**2**answers

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**

votes

**1**answer

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

**2**answers

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

**4**answers

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 $n$-...

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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

**4**answers

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 ...