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

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0answers
22 views

Why having a conserving metric defined on a product space guarantees that projections are isometries? [on hold]

I wanted to know when the metric (between 2 spaces A and B) is preserved. Is metrics preserved when isometries are projections that guarantee or define a product space existence within or of product ...
-3
votes
2answers
69 views

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

What is an example of a connected Hausdorff space $X$ with $|X|>1$ and a surjective continous map $f:X\to (X\times X)$?
7
votes
0answers
73 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 zeroth ...
3
votes
0answers
38 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 ...
2
votes
0answers
26 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 ...
3
votes
1answer
62 views

Is the category of inclusion prespectra bicomplete?

Working in compactly generated weak Hausdorff spaces, is the category of inclusion prespectra bicomplete? I should probably specify that by inclusion prespectra, I mean prespectra such that the ...
2
votes
0answers
73 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_{\...
3
votes
1answer
161 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 ...
0
votes
1answer
48 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 ...
3
votes
0answers
147 views
+200

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 $...
1
vote
1answer
47 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!
5
votes
0answers
206 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 ...
5
votes
2answers
200 views

Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?

Short version of question. Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ such that all points of $S$ have distinct pairwise distances? Formal version of question. If $X$ is a set, let $[X]...
4
votes
0answers
44 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[...
4
votes
1answer
193 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 $$\{...
1
vote
0answers
59 views

Explicit description of the scheme obtained by relative gluing data over a base scheme

I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
1
vote
1answer
42 views

A sequence in generalized order spaces

Let $X$ be a GO-space with the topology $\tau$ and $\lambda$ be the usual open interval topology on $X$. Put $$ R= \{x\in X: [x, \rightarrow) \in \tau\setminus \lambda \} \text{ and } L= \{x\in X: (\...
4
votes
1answer
100 views

Does every cut-point space embed into the plane?

Let $X$ be a connected separable metrizable topological space. Call it a cut-point space if $X\setminus \{x\}$ is disconnected for every $x\in X$. Then does $X$ embed into the plane? My thoughts: (...
4
votes
0answers
329 views

Questions about obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
3
votes
1answer
106 views

Continuity concepts for correspondences

Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F ...
1
vote
2answers
94 views

Extending homeomorphisms between compact metric subsets

Let $X$ be a compact metric, second countable space with finite covering dimension. Let $A,B$ be two closed subsets of $X$. Assume that $h:A\to B$ is a homeomorphism. Is it possible to extend $h$ to a ...
1
vote
0answers
79 views

How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...
1
vote
1answer
66 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, ...
3
votes
1answer
124 views

Is there a metaLindelof nonLindelof space which has a dense hereditarily Lindelof subspace?

My question is as the title, i.e., Is there a metaLindelof nonLindelof space which has a dense hereditarily Lindelof subspace? The question is related to the following result: Every separable ...
8
votes
4answers
645 views

How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]

Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
6
votes
1answer
228 views

“Cyclic” continuum

On p. 221 of http://topology.auburn.edu/tp/reprints/v08/tp08113.pdf, I found the following definition: "A curve is said to be cyclic if its first Čech cohomology group with integer coefficients ...
2
votes
1answer
71 views

Is the following map a cofibration?

Assuming that the diagonal map $X\rightarrow X\times X$ is a cofibration. Is it true that the diagonal map $\Sigma X\rightarrow \Sigma X\times \Sigma X$ is a cofibration? (Where $\Sigma X$ is the ...
8
votes
1answer
208 views

Intersection of nested open ball in complete metric spaces is nonempty?

My question is that whether the following statement is true or not. In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies $$ \exists \epsilon > ...
4
votes
1answer
173 views

Bounded growth of functions vs bounded growth of functions on countable sets

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean. Let $...
1
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0answers
98 views

Commutations of some limits and colimits in $\mathbf{CGWH}$

I know that finite limits do not commute with filtered colimits in general in $\mathbf{CGWH}$, nevertheless, do colimits commute with pullbacks, when we consider diagrams of the form $$\begin{matrix}&...
4
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0answers
58 views

An easier example of complete lattice such that the Scott topology on it is not sober

Basic notions: $1$, A partially ordered set is a dcpo if each of its directed subsets has a supremum. (https://en.wikipedia.org/wiki/Complete_partial_order)\ $2$, A subset O of a dcpo P is called ...
2
votes
1answer
64 views

A monoidal model structure on pointed spaces

Do the classes of pointed Hurewicz cofibrations, pointed Hurewicz fibrations and pointed homotopy equivalences give a model structure on pointed (compactly generated weak Hausdorff) topological spaces ...
0
votes
0answers
61 views

Reference sought: maximal ideal space of $\ell_\infty$ modulo sequences that go to 0 along a filter

It is not too hard to show the following. Suppose that $\mathcal{F}$ is a non-principal filter on $\mathbb N$. Denote by $c_0^{\mathcal{F}}$ the subspace of $\ell_\infty$ consisting of sequences that ...
2
votes
1answer
77 views

Are separability and ccc equivalent for closed subspaces of $\beta N$?

Let $\beta \mathbb N$ be the Stone-Cech compactification of the integers. Then $\beta \mathbb N\setminus \mathbb N$ is non-separable because if fails the ccc condition, that is, it has an uncountable ...
1
vote
1answer
200 views

Sufficient conditions for a topological space to be regular $T_3$

There was a similar thread on the neighbour forum StackExchange on sufficient conditions for a topological space to be completely regular $T_{3^1/_2}$. Please, let me know any known condition(s) that ...
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votes
1answer
86 views

connected set of sum of upper semi continuous function [closed]

Let $C(X)$:space of continuous functions on a compact space.Topology $C(X)$ is generated by sup-norm($||T||=sup_{v}\frac{||T(v)||}{||v||}$). Consider $f$ and $g :C(X)\rightarrow \mathbb{R}$ are upper ...
0
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0answers
7 views

Are the following sets closed/open on the subspace topology [migrated]

I'm a graduate student in mathematics whose taking a topology class. The class recently started and I've been studying like crazy trying to understand, adapt and learn methods that work for writing ...
3
votes
1answer
74 views

Reduced suspension of a Hurewicz cofibration

I would like to know whether the reduced suspension of a Hurewicz cofibration of pointed spaces (it is a Hurewicz cofibration when considered as a map of unbased spaces) is an acyclic Hurewicz ...
2
votes
1answer
76 views

Space which is $T_1$ and sober but not Hausdorff?

Every Hausdorff space is $T_1$ and sober. Does the converse hold? I expect not. What's a counterexample? I expected I should be able to look this up in Counterexamples in Topology, but unfortunately ...
4
votes
1answer
147 views

Embedding ordinals with the order topology into connected $T_2$-spaces

Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\...
1
vote
2answers
73 views

Can the Boolean Algebra of regular open sets be isomorphic to ${\cal P}(\omega)/(\text{fin})$?

Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...
4
votes
2answers
103 views

Inverse image of rational values

I am a postgraduate student of physics. While doing some research on Poincare's work on the integrability of the three body problem, I came up with the following problem (which I feel unable to handle,...
5
votes
2answers
169 views

Codimension-1 subgroups of 3-manifold groups

Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C_{G}/H$ has more than one end, where $C_{G}$ is the Cayley graph of $G$. Do all ...
3
votes
1answer
58 views

Is the compact-open topology on the dual of a separable Frechet space sequential?

Let $X$ be a separable Frechet space (= Polish locally convex linear metric space) and $X'_c$ be the space of linear continuous functionals on $X$, endowed with the compact-open topology (= the ...
1
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1answer
107 views

The completeness of spaces of continuous functions with the compact-open topology

For a Tychonoff space $X$ let $C_k(X)$ denote the space of continuous real-valued functions on $X$, endowed with the compact-open topology. Problem. Is the space $C_k(X)$ Polish if it is Polishable ...
1
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1answer
61 views

Why is a certain space of linear isometries paracompact

Let $V$ be a finite dimensional real inner product space and $U$ a real inner product space of countable dimension. Why is the space of linear isometries from $V$ to $U$ paracompact?
24
votes
3answers
691 views

Does $M^o=N^o$ imply that $\partial M = \partial N$?

let $M$ be a smooth $n$-manifold with boundary $\partial M$; I denote by $M^o$ the internal part of $M$, that is $M \smallsetminus \partial M$. The question is the same as in the title: let $M$ and $N$...
2
votes
1answer
86 views

Commutation of filtered colimits and finite limits in $\mathbb{CGWH}$

Do filtered colimits and finite limits (in particular pullbacks) commute in the category of compactly generated weak Hausdorff spaces?
0
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0answers
137 views

Curve such that $f(0)$ is in the interior of $f(]0,x[)$

Is there an example of continuous map $f:\mathbb R\to \mathbb R^2$ such that there exists $a\in \mathbb R$ such that for all $]y,z[\subset \mathbb R$ , $f([y,z])$ has non empty interior and such that ...
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votes
2answers
137 views

A countable polish space must be discrete? [closed]

I am looking for an elegant proof of the fact that a countable metric space is complete iff its underlying topology is discrete. It is easy to see that a discrete space is complete because its ...