# Questions tagged [gn.general-topology]

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

**1**

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**2**answers

116 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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**0**answers

56 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 ...

**-1**

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**0**answers

70 views

### How is a simplicial map a cellular map? [on hold]

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

**1**answer

84 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\...

**0**

votes

**0**answers

78 views

### Clarification about the ? -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 ...

**3**

votes

**0**answers

98 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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**0**answers

77 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 ...

**2**

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**2**answers

131 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 ...

**37**

votes

**2**answers

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**

votes

**2**answers

131 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**

votes

**1**answer

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 ...

**8**

votes

**2**answers

263 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**

votes

**1**answer

97 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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**0**answers

147 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**

votes

**0**answers

186 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 $\mathbb R^3$ ...