# Questions tagged [gn.general-topology]

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

3,050
questions

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### A Uniform Metric Selection Theorem

Let $X$ and $Y$ be bounded complete separable metric spaces. Let $C = 2^\omega$ be Cantor space with its standard metric. All product spaces are taken to have the max metric.
Let $F, G \subseteq X\...

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**2**answers

161 views

### Some special closed sets in the Bohr compactification of the reals

Let $X$ denote the Bohr compactification of the reals. What can be said about the intersection of $\overline{\mathbb R^+}^X$
with $\overline{\mathbb R^-}^X$, the closures in $X$ of $\mathbb R^+:=\{...

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35 views

### Non-cut points and metrizability

In a paper I'm writing, I happened upon a proof of:
Theorem. If $X$ is a separable Hausdorff continuum with only countably many non-cut points, then $X$ is a dendrite.
A continuum is a compact ...

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80 views

### Size of the orbit of a dense set

This question is a follow-up to: this post.
Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big ...

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161 views

### Haar-null union of dense subsets

Let $\{X_i\}_{i \in \mathbb{R}-\{0\}}$ be a set of subsets of a separable infinite-dimensional Fréchet space $X$ and $I$ be uncountable. Moreover, suppose that
(Dense $G_{\delta}$) $X_i$ is a dense ...

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58 views

### The Borel class of a countable union of $G_\delta$-sets, which are absolute $F_{\sigma\delta}$

Problem. Assume that a metrizable separable space $X$ is the countable union $X=\bigcup_{n\in\omega}X_n$ of pairwise disjoint $G_\delta$-sets $X_n$ in $X$ such that each $X_n$ is an absolute $F_{\...

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86 views

### Generalization of pseudogroups

Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup
One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between ...

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128 views

### Locally compact cut-point space homeomorphic to reals?

Suppose $X$ is a connected separable metric space, and $X\setminus \{x\}$ has exactly two connected components for every $x\in X$.
If $X$ is locally connected, then $X\simeq \mathbb R$. This was ...

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51 views

### “Robust” Noninjectivity of a Continuous Mapping of a Sphere into the Plane

Let $X=\mathbb{S}^2$ and $Y=\mathbb{R}^2$ and $f:X\to Y$ a continuous mapping. Is it true that there must exist a nonempty set $V\subset f(X)$, open in $f(X)$ (in the subspace topology), such that ...

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199 views

### Ideal characterization of almost convergence

$\bullet$ A real sequence $x=(x_n)_n$ is called convergent to $\alpha$ in usual sense if for any $\epsilon>0$ the set $\{n\in\mathbb N:|x_n-\alpha|\geq\epsilon\}$ is finite.
$\bullet$ A real ...

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56 views

### Embedding of regular spaces in countably compact Hausdorff spaces

It is well-known that each completely regular space embeds into a compact Hausdorff space.
Problem. Is it true that each regular space embeds into a countably compact Hausdorff space?
The problem ...

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**1**answer

161 views

### Functor From Rings into Compact Hausdorff Spaces

There is an adjunction $\text{Bool}^{op} \leftrightarrow \text{Set}$ between boolean algebras and sets which sends a boolean algebra to the set of its prime ideals and a set $X$ to $[X, \mathbb{F}_2]_{...

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62 views

### Rough classification of Peano Curves

By Peano curve I mean a continuous map from the unit interval that fills the unit square in $\mathbb R^2$.
In the paper:
Shchepin, E. V.; Bauman, K. E., Minimal Peano curve, Proc. Steklov Inst. Math....

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87 views

### Hilbert Scale Inclusions

I'm looking at properties of the scale of Hilbert spaces $(X_s)_{s\in \mathbb{R}}$, which are constructed as follows. Starting with $A:D(A)\subset H\to H$, $A$ a densely defined, strictly positive ($...

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82 views

### Subspaces of compact spaces and quotients of Hausdorff spaces

Let $\operatorname{Top}$ be the class of topological spaces. Furthermore, let $\mathcal{U}\subset\operatorname{Top}$ and $\mathcal{V}\subset\operatorname{Top}$ classes satisfying the following ...

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155 views

### Is a free and discrete group action on the plane a covering space action?

Let $\mathbb{R}^2$ be the plane, and let a group $G$ act on it with orientation preserving homeomorphisms, and assume that
every orbit of $G$ is a discrete subset in $\mathbb{R}^2$
$G$ acts freely: ...

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**1**answer

2k views

### Which polygons can be turned inside out by a smooth deformation?

Take a non-degenerate polygon with side lengths $\{a_1,\dots,a_n\}$ in a convex configuration. What is the condition on the $a_i$'s so that the polygon can be turned inside out by a continuous motion ...

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469 views

### Open subspaces of CW complexes

I am looking at the paper
Covering homotopy properties of maps between CW complexes or ANRs
by
Mark Steinberger and James West
and a claim is made in the proof of their first main theorem ...

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116 views

### Subsets of reals which are both $F_{\sigma\delta}$ and $G_{\delta\sigma}$

Let $X\subseteq \mathbb R$ such that
$X$ is an $F_{\sigma\delta}$-set (in $\mathbb R$); and
$X$ is a $G_{\delta\sigma}$-set.
It is not necessarily true that $X$ must be $F_\sigma$ or $G_\delta$. A ...

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139 views

### topological properties of $G_{\delta}$ sets in a compact Hausdorff space

I am trying to understand a family of types $\mathcal{F}$ in the set $S(A)$,the set of complete types over $A$ (in the sense of types in model theory) which is a compact and Haurdorff space equipped ...

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90 views

### Collineations of projective spaces and isomorphisms of fields

For a (topological) field $F$ by $FP^2$ we denote the projective plane, i.e., the quotient space of $F^3\setminus\{0\}^3$ by the equivalence relation $\vec x\sim\vec y$ iff $\vec x=\lambda\vec y$ for ...

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93 views

### Properties of the topology of sequential convergence $\tau_{seq}$

Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_{seq}$ has the following ...

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160 views

### On the universal property for interval objects

In his lecture, The Categorical Origins of Lebesgue Measure, Professor Tom Leinster mentions the following theorem:
Theorem 1: (Freyd; Leinster) The topological space $[0, 1]$ comes equipped with ...

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258 views

### Does using continued fractions work to give a homeomorphism $\mathbb{Q}^+ \rightarrow (\mathbb{Q}^+)^2$?

Let $\mathbb{Q}$ be the topological space of rational numbers (with topology induced by inclusion in the real line) and let $\mathbb{Q}^+$ be the set of positive ($x>0$) rationals.
I'm looking ...

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61 views

### If all length metrics are strong equivalent on a closed connected topology manifold?

Let $M$ be a connected closed topology manifold and $d$ is a length metric (or an inner metric) on it , i.e. the distance between every pair of points is equal to the infimum of the length of ...

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60 views

### Structure of extensions arising in Lie approximation of connected groups

My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...

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96 views

### Cardinal Invariants and Physics

There are many applications of topology to physics, but I wonder if there is a known application of cardinal invariants to physics.

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148 views

### Must a locally compact, second countable, Hausdorff space support a Radon measure?

Let $X$ be a locally compact, second countable and Hausdorff space, must there be a Radon measure on $X$ whose support is $X$?
The motivation for this question comes from Anton Deitmar's paper On ...

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276 views

### In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point?

I previously asked In which topological spaces does the existence of a loop not contractable to a point imply there is a non-contractable simple loop also?
Given the broad scope of this question I ...

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197 views

### In which topological spaces does the existence of a loop not contractible to a point imply there is a non-contractible simple loop also?

In another MathOverflow post I asked: In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
Note that ...

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138 views

### Is every closed subset of finite measure contained in an open subset of finite measure?

Could someone will verify my statement: For every locally finite Borel measure on metric space and closed set $F$ with finite measure, there exists open set $U$ such that $F \subset U$ and $U$ has ...

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4k views

### In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?

I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces.
If it is true that:
In a Topological Space, if there exists a loop that cannot ...

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148 views

### Is this set of fractions dense in the interval $\big[\frac 13,\frac 12\big)$?

I have an interest in the set
$$A= \bigg\{\frac{ab+c}{(2a+1)b+c}\,\bigg|\, a \in {\mathbb Z}^+, b\in{\mathbb Z}^+~\text{is \((a+1)\)-smooth}, 0\leq c\leq ab\bigg\}.$$ In particular, is $A$ dense in ...

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348 views

### Is the set of escaping endpoints for $e^z-2$ completely metrizable?

Let $f:\mathbb C \to \mathbb C$ be the complex exponential $$f(z)=e^z-2.$$ It is known that $J(f)$, the Julia set of $f$, is a uncountable collection of disjoint rays (one-to-one continuous images ...

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35 views

### Sobolev tensor spaces and finite ranks

Let $W^{2,2}(\Omega_i)$, $\Omega_i = [-1,1]$, $i = 1,\ldots,d$ be the Sobolev spaces of twice weakly differentiable, square integrable functions. Let further $\otimes_a$ denote the algebraic tensor ...

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325 views

### Normal bundle of Whitney embedding

Let $X$ be a real $n$ dimensional manifold. One knows that it can be embedded into $\mathbb{R}^{2n}$ by the Whitney embedding theorem. The normal bundle for such an embedding will be a rank $n$ real ...

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33 views

### Is the space of $p$-geometric rough paths is Homeomorphic to Frechet Space

Let $\Omega G^p([0,T];\mathbb{R}^n)$ be a space of $p$-geometric rough paths with values in $\mathbb{R}^n$. Is $\Omega G^p([0,T];\mathbb{R}^n)$ homeomorphic to some Fr\'{e}chet space?

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53 views

### Are $T_0$ topological quasigroups completely regular?

In 1957 H. Salzmann generalized to quasigroups but weakened the standard result that $T_0$ topological groups are completely regular. He was able to show that $T_0$ topological quasigroups are regular ...

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44 views

### $L^p $ Space with Values in Metric Space Homeomorphic

Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be complete $\lambda$-doubling metric measure spaces and $p \in [1,\infty)$. Moreover, suppose that there exists a homeomorphism $\Phi$ from $(Y,d_Y)$ to some ...

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857 views

### Link of a singularity

I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$.
If we set $x = x_1+ix_2, y = y_1+iy_2, z ...

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171 views

### Research in compactifications of locally compact spaces

I would like to know how is it going the research in compactifications of locally compact Hausdorff spaces. Are there people doing this? Are there relevant conjectures on it?

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171 views

### The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces

Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g.
N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from
https://neil-strickland.staff.shef.ac.uk/courses/...

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162 views

### Do the higher levels of the Borel hierarchy correspond to absolute topological properties?

It is well known that a subset $Y$ of a Polish space $X$ is completely metrisable iff it is a $G_\delta$ subset. This relates a relative topological property of the subspace $Y \subset X$ to an ...

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227 views

### Is a cofinite topology for a set with cardinality between $\aleph_{0}$ and $2^{\aleph_{0}}$ path-connected?

It is easy to show that $\mathbb{N}$ with the cofinite topology is not path connected and that any set with cardinality $\geq 2^{\aleph_0}$ equipped with the cofinite topology is in fact path ...

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140 views

### Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]

Please let me denote the following
(TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold
(PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF
(PL) ...

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152 views

### Fibers of continuous maps of $\mathbb{R}^n$ which are injective at dense points

Question. Suppose that $f\colon\mathbb{R}^n \to \mathbb{R}^n$ is a continuous map and there is a dense subset $D \subset \mathbb{R}^n$ such that $f^{-1}(f(x)) = \{x\}$ for all $x \in D$.
Is every ...

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131 views

### Equivalence of $\sigma$-weak topology to another topology

Let $\mathcal H$ be a Hilbert space. Define a topology $\tau_1$ on $B(\mathcal H)$ by the family of seminorms $x\mapsto |Tr(xa)|,$ $a\in L^1(B(\mathcal H)).$ Here $B(\mathcal H)$ denotes the set of ...

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218 views

### subspace topology and strong topology

Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...

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208 views

### Properties of Frechet distance

There is an old construction, apparently due to Frechet's PhD thesis (which is unfortunately written in French and in ancient notation), which turns the set of curves in a metric space modulo ...

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725 views

### Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.
Cobordism Theory for DIFF/Differentiable/smooth manifolds
However, there are ...