# 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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### Do manifolds inherit topological proporties of their models?

Suppose a space is for example paracompact, then a manifold that modeled on it is paracompact? It seems that it is true but how we can show it. More general questions what topological properties ...

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45 views

### Fully faithful functor from schemes to spaces

Is there a fully faithful functor from the category of schemes
to the category of topological spaces and continuous maps (or some other sufficiently topological objects, e.g. smooth manifolds and ...

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53 views

### Are compactifications of completely $T_{4}$ spaces completely $T_{4}$?

The title is the question.
Given a locally compact completely $T_{4}$ space $X$ (every subspace is $T_{4}$) and a (Hausdorff) compactification $\overline{X}$ of $X$, is $\overline{X}$ also completely ...

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81 views

### Clopen base for topological space [on hold]

Let $(X,d)$ a metric space, $B(x,\varepsilon):=\{y\in X|~ d(x,y)<\varepsilon\}$ where $x\in X$ and $\varepsilon>0$ and $(X,\mathcal{T}_d)$ the natural topological space for the metric space $(X,...

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42 views

### Closed union of all connected subsets that contain x [on hold]

If Cx(S) is the union of all connected subsets of S which contain x, it is connected. I understand that, but what I don’t understand is that if S is closed, then Cx(S) is closed. Isn’t that like ...

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46 views

### Can closure and complement generate 14 distinct operations on a topological space if no subset generates more than 6 distinct sets?

$\newcommand{\XT}{(X,\mathcal{T})}$From Definition 1.2 in Gardner and Jackson (GJ): The $K$‑number $K(\XT)$ of a topological space $\XT$ is the cardinality of the Kuratowski monoid of operators ...

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47 views

### A retract algebraic subset of the plane which does not admit an algebraic retraction

What is an example of an algebraic (=Zariski closed) subset $C$ of $\mathbb{R}^2$ which is a topological retract of $\mathbb{R}^2$, but there is no algebraic retraction $P:\mathbb{R}^2 \to C$?
What ...

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94 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 the countably infinite product space equipped with the product topology. Let $\mathcal A$ be any field of ...

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103 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 ...

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168 views

### A simple proof of Jordan curve theorem [closed]

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

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137 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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47 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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142 views

### Requirement for connected sets [closed]

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 ...

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64 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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95 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\...