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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.

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11 views

On the connected sum of a surface with a torus

I am studying the classification of Surfaces, and run into the notion of connected sum. We define it in terms of triangulations. I want to show the following. Let $S$ be a triangulated surface. I want ...
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0answers
23 views

If the finitely additive measure of an open set is approximable by clopen sets, is it approximable from within?

Let $F$ be a finite set equipped the discrete topology. Let $X = F \times F \times ...$ be equipped with the product topology. Let $\mathcal A$ be any field of subsets of $X$ that contains the open ...
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0answers
34 views

Does each $\omega$-narrow topological group have countable discrete cellularity?

A topological space $X$ is defined to have countable discrete cellularity if each discrete family of open subsets of $X$ is at most countable. A family $\mathcal F$ of subsets of a topological space ...
4
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0answers
156 views

A simple proof of Jordan curve theorem [on hold]

I need a short proff of the Jordan curve theorem please. The one I have is 16 pages long and is for a little expo, so I need one a little shorter. Thanks
3
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0answers
119 views

Topological Singularities in Affine Varieties

Let $X$ be an affine variety over $\mathbb{C}$. Let $x\in X$. If $x$ is non-singular, then $x$ is locally holomorphic (in the Euclidean topology). See here for a relevant MO post. By results of ...
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0answers
36 views

Does the total space of a bundle satisfy the Tietze extension property when the fiber and base space do satisfy this property?

We say that a Topological space $Y$ satisfies the Tietze extension propery, TE property, if in the formulation of Tietze extension theorem "$\mathbb{R}$" can be replaced by $Y$. Obvioysly the ...
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2answers
142 views

Requirement for connected sets [on hold]

Let $E$ be a compact metric space. Suppose that closure of every open ball $B(a,r)$ is the closed ball $B'(a,r)$. Must every open ball in $E$ be connected? I think it most probably is. But I don't ...
3
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0answers
60 views

Is there a T3½ category analogue of the density topology?

Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology ([1]) but for category (and meager sets) instead of ...
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0answers
70 views

How is a simplicial map a cellular map? [closed]

I’m fairly new to studying topology and I’m currently reviewing cellular maps, can someone explain how a simplicial map between geometric complexes is also a cellular map?
4
votes
1answer
85 views

Reference request: filter tends to filter along map

Recall that a filter on a set $X$ is a nonempty collection $\mathcal{F}$ of subsets of $X$ such that (i) $U\subseteq V\subseteq X$ and $U\in\mathcal{F}$ implies $V\in\mathcal{F}$, and (ii) $U,V\in\...
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0answers
82 views

Clarification about the ϵ -net argument

I have been reading the paper Do GANs learn the distribution? Some theory and empirics. In Corollary D.1, they reference the paper Generalization and Equilibrium in Generative Adversarial Nets which ...
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0answers
103 views

Continuous open self maps on Cantor space

A continuous self map on the Cantor space $C = \{0,1\}^\mathbb{N}$ is a mapping $f = (f_i)_{i\in \mathbb{N}}$ such that each $f_i$ is a map from $C$ to $\{0,1\}$ that depends only on a finite number ...
4
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0answers
80 views

Universal and strong $Q$-sets

A subset $X\subset \mathbb R$ is called $\bullet$ a $Q$-set if for every subset $A\subset X$ there exists a $\sigma$-compact set $C\subset\mathbb R$ such that $C\cap X=A$; $\bullet$ a strong $Q$-set ...
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2answers
134 views

The space of Borel function modulo comeager sets is Dedekind complete

Let $(X,\tau)$ be a topological space. Denote by $Bor(X)$ the space of Borel functions $f:X\to\mathbb{R}$ where we identify two functions whenever they agree on the complement of a meager set. We ...
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2answers
1k views

Ultrafilters as a double dual

Given a set $X$, let $\beta X$ denote the set of ultrafilters. The following theorems are known: $X$ canonically embeds into $\beta X$ (by taking principal ultrafilters); If $X$ is finite, then there ...
3
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2answers
134 views

Ultraweak topology in abelian von Neumann algebras

Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff ...
3
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1answer
127 views

Nonexistence of a 'product universal' compact Hausdorff pseudotopological space?

I'm largely following the definitions of this paper, but I will replicate the relevant ones here. I'm taking a pseudotopological space to be a set $X$ together with a relation $\rightarrow$ on the ...
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2answers
270 views

Small uncountable cardinals related to $\sigma$-continuity

A function $f:X\to Y$ is defined to be $\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
0
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1answer
99 views

Reference request: Baire class 2 functions

There are many articles on Baire 1 functions, but not many on Baire 2 and above. Where can I find a nice comprehensive survey of them?
3
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0answers
155 views

A user guide to the theory on Corks

I am trying to digest the meanings of the corks from the both: algebraic topology and geometry topology perspectives. Studying corks is important for understanding the exotic phenomenon of 4-...
17
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0answers
188 views

Can 4-space be partitioned into Klein bottles?

It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ ...
2
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2answers
115 views

Topological dimension of the image of continuous surjective functions

Consider two topological spaces $(X,\tau)$ and $(Y,\omega)$ and a continuous surjective function $f\colon X\to Y$. Let $\mathrm{dim}(X)$ and $\mathrm{dim}(Y)$ denote the Lebesgue covering dimension ...
2
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1answer
69 views

$T_1$-spaces vs $T_1$-hypergraphs

Let us say that a hypergraph $H=(V,E)$ is $T_1$ if for $x\neq y$ there is $e\in E$ such that $e\cap\{x,y\} = \{x\}$. Note that for any $T_1$-space $(X,\tau)$ the topology $\tau$ contains the ...
2
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1answer
208 views

Scattered separators in Erdős space

Let $X$ be the set of all points in $\ell^2$ with all rational coordinates. $X$ is known to be totally disconnected, but $X$ is not zero-dimensional. For instance, the empty set does not separate the ...
3
votes
1answer
180 views

Extending a continuous self-map of an open subset to the whole space

We say that a $T_2$-space has the open extension property (OEP) if for any open set $U$ and continuous map $f:U\to U$ there is a continous map $g:X\to X$ such that $g|_U = f$. The space $\mathbb{R}$ ...
1
vote
1answer
195 views

Hairy ball theorem for odd-dimensional spheres

Let $\mathbb S^n$ be the $n$-sphere: $$\mathbb S^n=\left\{x \in \mathbb R^{n+1}: \left\|x\right\|=1\right\}.$$The hairy ball theorem can be formulated as follows: If $n$ is even and $f\,\colon\, \...
1
vote
1answer
136 views

$n$-product-periodic topological spaces

We call an topological space $(X,\tau)$ $n$-product-periodic for an integer $n\geq 3$ if $\prod_{i=1}^n X \cong X$ but for all integers $k$ with $2\leq k\leq n-1$ we have $\prod_{i=1}^k X \not\cong X$....
2
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2answers
211 views

Are there $2^{\aleph_0}$ pairwise non-isomorphic Boolean algebras on $\omega$?

Is there a collection of $2^{\aleph_0}$ pairwise non-isomorphic countable Boolean algebras? Equivalently, are there $2^{\aleph_0}$ pairwise non-homeomorphic closed subsets in the Cantor space?
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0answers
61 views

Norm closure of $C_b^1(\mathbb{R})$

I want to determine what the closure of $C_b^1(\mathbb{R})$, the space of continuous differentiable functions with bounded derivative, with respect to the supremums norm is. I think that $\overline{...
4
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1answer
155 views

Contractible $T_2$-space without subspace homeomorphic to it

This is an update to an older question. Is there a contractible $T_2$-space $(X,\tau)$ on more than $1$ point such that no proper subspace of $X$ is homeomorphic to $X$?
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0answers
139 views

A special connected subset of the Cantor fan

Is there a dense connected subset $X$ of the Cantor fan $$(C\times [0,1])/(C\times \{1\})$$ such that for every two connected subsets $X_1,X_2\subseteq X$, the intersection $X_1\cap X_2$ is connected? ...
12
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1answer
392 views

Are Hausdorff measures on the real line Haar measures for some locally compact topology?

For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...
3
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2answers
138 views

Rigid space, but with homeomorphic neighborhoods

What is an example of a topological space $(X,\tau)$ on more than one point, with the following properties? the only homeomorphism from $X$ to itself is the identity, and given $x,y\in X$ there are ...
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1answer
46 views

Two notions of boundedness in metrizable topological vector space [closed]

In a metrizable topological vector space X with the metric d, a subset A is said to be bounded if it can be absorbed by any neighbourhood of 0 and a subset A is said to be d-bounded if its diameter ...
1
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1answer
85 views

Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive

Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty. Let $S : X → X$ and ...
2
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0answers
40 views

Metrically homogeneous spaces as inverse limits

Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following: Is $X$ ...
1
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0answers
68 views

Minimal size of an open affine cover for an open complement

Let $X$ be a smooth projective scheme and $Y$ be a projective subscheme of $X$, not necessarily smooth. Are there any known results about the minimal size of an open affine cover (number of affines in ...
5
votes
1answer
99 views

Does every bijective graph endomorphism restrict to a full-cardinality isomorphism?

Given a graph $G$, and a bijective endomorphism $f$ (that is, a graph homeomorphism $f : G \to G$ that establishes a bijection on the vertices), it is true that $f$ is an automorphism whenever $|G|$ ...
4
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2answers
212 views

Set of topologies on a group making it a compact Hausdorff topological group

Maybe stupid, but from the following well known facts about compact Hausdorff (CH) spaces: CH topologies on a given set are pairwise incomparible (one is not finer or coarser than the other). There ...
1
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1answer
61 views

Basis or subbasis for Scott topology

Let $X$ be a partially ordered set. A subset $S\subseteq X$ is called Scott-open if and only if it is: Upward-closed: $x\in S$ and $x\le y$ implies $y\in S$; Inaccessible by directed suprema: if $D\...
0
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1answer
51 views

Empty interior lack of minima

Suppose that $U \subseteq \mathbb{R}^d$, and satsifies $U$ is dense in $\mathbb{R}^d$, U has empty interior, Then is it possible that $$ \inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x), $$ ...
2
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1answer
45 views

Is the Scott topology generated by the ideals as the closed sets?

Let $X$ be a directed-complete partial order, or even a complete lattice. A subset $S\subseteq X$ is called Scott-closed if and only if it is: Downward-closed: $y\in S$ and $x\le y$ implies $x\in S$; ...
6
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1answer
140 views

Geometry of complements to compacts of codimension 2

Let $K\subset \mathbb{R}^n$ be a (nonempty) compact of covering dimension $\le n-2$. In particular, $K$ does not separate $\mathbb{R}^n$ (even locally). I will equip $M=\mathbb{R}^n-K$ with the ...
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0answers
28 views

Localized connected expansions

Given a connected space, it is easy to tell if there is a connected expansion because maximal connected spaces (those admitting no finer connected topology) have the property that every dense subset ...
3
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0answers
44 views

Galois Covering induces new Cover $Ind_H ^G(Y)$

I have a question about the construction of the so called "induced cover" introduced in Tamas Szamuely's "Galois Groups and Fundamental Groups" (see page 84): We consider a group $G$ which contains a ...
2
votes
1answer
90 views

Non-homeomorphic computable metric spaces whose computable points are computably homeomorphic

This is a follow-up of sorts to an earlier question on mine in that it should be easier to construct a positive example of this, if it exists. To be clear about definitions, a computable metric space ...
1
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1answer
112 views

Separability of the Stone space of a free sigma-algebra

Let $X$ be the Stone space of the free $\sigma$-algebra $A$ on $\omega_1$ free generators. Is $X$ separable (i.e. does $X$ contain a countable dense set)?
8
votes
1answer
182 views

Measure support decomposition that “tends to infinity”

I would like to know the answers to the following two questions. Let $S$ be a locally compact Hausdorff space, $\mu$ be a regular Borel measure with non-compact support $M$. Denote $$ \mathscr{H}=\{\...
6
votes
1answer
243 views

Covering compact Hausdorff spaces with closed $G_\delta$ sets

I'm thinking about results of the form: Under assumption $A$, if $X$ is a compact Hausdorff space and $C$ is a cover of $X$ by closed $G_\delta$ sets, then there is a subcover of cardinality $\leq\...
10
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0answers
194 views

Planar arc on a topologically embedded sphere or disk in $\mathbb{R}^3$

An arc is a set homeomorphic to the unit interval $[0,1]$; an arc in $\mathbb{R}^3$ is planar if it is contained in some plane. The following questions are motivated by Anton Petrunin's Disc bounded ...