# 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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### If $X$ has the fixed point property, what about $\text{Cont}(X,X)$?

If $(X,\tau)$ is a topological space, we denote by $\text{Cont}(X,X)$ the collection of all continous functions $f:X\to X$. We say that $(X,\tau)$ has the fixed point property if for any $f\in\text{...

**1**

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**0**answers

75 views

### Cobordism of an annulus with a non-vanishing vector field

Let $M$ be a compact three-dimensional manifold with corners, which is a cobordism of the two-dimensional annulus. In particular, the codimension one boundary of $M$ consists of two copies of the ...

**4**

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**1**answer

172 views

### Does the notion of a compactly generated space (or $k$-space) depend on the choice of universe?

We recall the notion of a $k$-space (or compactly generated space) to fix our notations. For every topological space $X$, we can define a category $\mathfrak{M}_X$. The class of objects of $\mathfrak{...

**11**

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**1**answer

222 views

### The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?

Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...

**9**

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**1**answer

310 views

### Noetherian spectral space comes from noetherian ring?

Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $\textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ...

**6**

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**1**answer

602 views

### A ridiculous combinatorial cardinal characteristic of the continuum?

This question assumes familiarity with combinatorial cardinal characteristics of the continuum. It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the ...

**5**

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**0**answers

116 views

### For which topological spaces does pullback along $\operatorname{ev}_0:B^I\to B$ have a right adjoint?

Let $B$ be a topological space. Consider the evaluation at zero of paths in $B$. This is a continuous map $\operatorname{ev}_0:B^I\to B$ where the domain carries the compact-open topology.
For which ...

**0**

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**1**answer

61 views

### Cardinality of the topology in countable connected $T_2$-spaces

If $(X,\tau)$ is a connected $T_2$-space with $|X|=\aleph_0$, what values can $|\tau|$ take?

**22**

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447 views

### Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?

On page 205 of his Topology book, James Munkres makes an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...

**-1**

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**1**answer

98 views

### Injective choice function for non-separable $T_2$-spaces

For any set $X$ and cardinal $\kappa$ let $[X]^\kappa$ be the collection of all subsets of $X$ of cardinality $\kappa$.
I was looking for $T_2$-spaces $(X,\tau)$ with the property that
$(P)$ ...

**3**

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**2**answers

136 views

### Is every uncountable, homogeneous connected $T_2$-space isomorphic to a subspace of $\mathbb{R}^\omega$?

We say a space $(X,\tau)$ is homogeneous if for any $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$.
What is an example of a connected, homogeneous $T_2$-space $(X,\...

**5**

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**1**answer

123 views

### Is the Euclidean topology on $\mathbb{R}$ contained in a maximal connected topology?

If $(X,\tau)$ is a connected space, then $\tau$ need not be contained in a maximal connected topology.
Is the Euclidean topology on $\mathbb{R}$ contained in a maximal connected topology?

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vote

**1**answer

85 views

### Understanding equivalent condition for covering dimension

Let dim $X$ denote the Lebesgue covering dimension for a topological space $X$. Now a result in common books concerning dimension theory states the following:
If $X$ is a normal topological space, ...

**2**

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**1**answer

140 views

### Homeomorphic open sets and total disconnectedness

If $(X,\tau)$ is a $T_2$-space such that all non-empty open sets are homeomorphic (with the subspace topology) to $X$, is there for all $x,y\in X$ with $x\neq y$ a clopen (closed and open) set ...

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**1**answer

87 views

### The preimage of continuum on Torus

Let $p:\mathbb{R}^2\rightarrow\mathbb{R}^2/\mathbb{Z}^2$ be the natural projection, obviously $\mathbb{R}^2/\mathbb{Z}^2$ is the torus $\mathbb{T}^2$, if $K$ is a connected and compact subset of $\...

**4**

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**1**answer

97 views

### $T_2$-spaces in which no two open sets are homeomorphic

This question was about spaces in which all non-empty open sets "look alike".
Now I am interested in the opposite: Is there a $T_2$-space $(X,\tau)$ with $|X|>1$ such that whenever $U\neq V$ are ...

**1**

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**1**answer

63 views

### Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...

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**6**answers

2k views

### Spectra of $C^*$ algebras

Gelfand-Naimark structure theorem for $C^* $ algebras gives a canonical isometric * isomorphism between any commutative unital $C^* $ algebra $A$ and the algebra of continuous complex-valued functions ...

**0**

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**1**answer

316 views

### The functor of continuous functions from compact CW-spaces to the reals

The contravariant functor $C(-)$ given by
$$
\hom_{Top}(-,\mathbb{R}):cCW\to Rng
$$
where $cCW$ is the category of compact CW complexes is injective on objects. What is known about surjectivity, ...

**13**

votes

**2**answers

935 views

### Order homomorphism functions on $\omega_1$

This question has now been published in a math journal, see update at the bottom.
I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains ...

**2**

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**1**answer

157 views

### effectively distinguishing knots

It was proven, I think by Mijatović EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound,...

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84 views

### Is each Peano continuum a topological fractal?

Problem. Is each Peano continuum a topological fractal?
A compact Hausdorff space $X$ is a topological fractal if $X=\bigcup_{i=1}^n f_i(X)$ for some continuous maps $f_1,\dots,f_n:X\to X$ such that ...

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107 views

### Completely I-non-measurable unions in Polish spaces

Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...

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74 views

### A community effort: equilibrium in quitting games [closed]

This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...

**5**

votes

**1**answer

164 views

### Finally dense implies dense

I am reading the article "A convenient category for directed homotopy" by Fajstrup and Rosicky and I have a doubt about the proof of Proposition 3.5. The setting is the following:
let $\cal{C}$ be a ...

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**1**answer

67 views

### sequences of iterated orthogonals (lifting property) in a category

I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property.
For example, several iterated orthogonals of $ \emptyset\...

**4**

votes

**1**answer

196 views

### Is each Swiatkowski function with closed graph continuous?

A function $f:\mathbb R\to\mathbb R$ is called Świątkowski if for any connected subset $C\subset \mathbb R$ and points $a,b\in C$ with $f(a)<f(b)$ there exists a continuity point $x\in C\setminus\{...

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126 views

### A ZFC-example of a countably compact paratopological group which is not a topological group

Problem. Does there exist a ZFC-example of a countably compact Hausdorff paratopological group which is not a topological group?
(The problem posed 27 May 2015 by Alexander Ravsky on page 9 of Volume ...

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337 views

### Baire category of tall ideals

Problem. Is it consistent with ZFC that $\mathfrak t=\omega_1$ and each $\omega_1$-generated tall $P$-ideal is of the second Baire category?
(Asked 01.10.2016 by David Chodounsky at page 20 of Volume ...

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**1**answer

51 views

### Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

This is a cross-post to the question I asked at MSE.
The set of Schur stable matrices is
\begin{align*}
\mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\},
\end{align*}
where $\rho(\cdot)$ denotes ...

**2**

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56 views

### Topological Shape Operator More Sensitive than Inverse Limits

This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...

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**1**answer

306 views

### How to prove that there does not exist any plane isotopy from the logarithmic spiral onto the real line? [closed]

Questions.
EDIT: readers please note that while this question arose in research, the OP was so hung-up on a question concerning infinite planar graphs that a strong a-forteriori-reason, kindly ...

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**3**answers

1k views

### Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology

The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be represented using ultrafilters.
Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ ...

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**1**answer

239 views

### Can we inductively define Wadge-well-foundedness?

For a topological space $X$ (which I'll identify with its underlying set of points), we define the Wadge preorder $Wadge(X)$: elements of the preorder are subsets of $X$, and the ordering is given by $...

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**1**answer

73 views

### Is the topology generated by the union of a chain of paracompact topologies paracompact?

Let $X$ be a set and let ${\frak T}$ be a collection of paracompact topologies on $X$ such that for any $\tau, \tau'\in {\frak T}$ we have $\tau\subseteq \tau'$ or $\tau'\subseteq \tau$. Let $\sigma$ ...

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**1**answer

241 views

### Generalizing the $T_0$-axiom

The starting point of this question is a slight reformulation of the $T_0$ separation axiom: A topological space $(X,\tau)$ is $T_0$ if for all $x\neq y\in X$ there is a set $U\in \tau$ such that $$\{...

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**1**answer

120 views

### The Stone-Čech compactification of the fixed point set

Let $G$ be a discrete group and $X$ be a Tychonoff $G$-space. Then there
exists a $G$-action on Stone-Čech compactification $\beta X$. If the
fixed point set $X^{G}\neq \emptyset $, then the Stone-...

**1**

vote

**1**answer

151 views

### Is every paracompact topology contained in a maximal paracompact topology?

If $(X,\tau)$ is a paracompact, is there a topology $\tau'\supseteq \tau$ such that $(X,\tau')$ is still paracompact, and $\tau'$ is maximal with respect to $\subseteq$ and paracompactness?

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101 views

### Example of connected Hausdorff space $X$ and surjective continous map $f:X\to X\times X$ [closed]

What is an example of a connected Hausdorff space $X$ with $|X|>1$ and a surjective continous map $f:X\to (X\times X)$?

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41 views

### Maps having the right lifting property against cofibrations of compact spaces

I would like to know the properties of the maps that have the right lifting property against cofibrations of compact spaces. By definition, they are acyclic Serre fibrations, but I would hope to be ...

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**1**answer

175 views

### Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ the conjuagation action fixes the first column?

Let $A \in GL_n(\mathbb R)$ be fixed. Let us consider the conjugation action by $G \in GL_n(\mathbb R)$, i.e., $G^{-1}AG$. I would like to see a way to identify the matrices such that the action fixes ...

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79 views

### Find a certain triangulation subordinate to a given covering of a manifold

Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\...

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**1**answer

706 views

### Interpreting a space in Baire space: how many facts do I need to understand the whole thing?

Below I'm working in ZF+DC+AD or similar; I want enough choice that things don't explode, but I also want the Wadge hierarchy to be well-behaved everywhere. Since this question is a bit long, I've put ...

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**1**answer

53 views

### A question on monotonically normal spaces

This question is related to one of previous questions.
For any generalized order space $X$, $X$ has countable tightness iff $X$ is first countable.
Since a generalized order space is monotonically ...

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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 ...

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**0**answers

137 views

### When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is a cross-post to the question I asked at MSE.
Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...

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**1**answer

49 views

### Is there a generalized order space $X$ with countable tightness which is not first countable?

I have a question concerning generalized order spaces.
Is there a generalized order space $X$ with countable tightness which is not first countable?
thanks a lot!

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**4**answers

637 views

### What was Burroni's sketch for topological spaces?

In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...

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237 views

### Grothendieck letter to Jun-Ichi Yamashita on tame topology

I am looking for Grothendieck writings on tame topology:
a manuscript on tame topology mentioned by Scharlau; a letter to Jun-Ichi Yamashita; a letter to Z.Mebkhout.
I am also interested in ...

**4**

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**0**answers

45 views

### The normality of powers versus the normality hypersymmetric powers

Let $X$ be a topological space. Let $[X]^{<\omega}$ be the space of non-empty finite subsets of $X$, endowed with the Vietoris topology. For a natural number $n$ the subspace $$[X]^{\le n}:=\{A\in[...