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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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      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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      1answer
      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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      votes
      1answer
      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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      0answers
      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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      1answer
      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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      vote
      1answer
      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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      votes
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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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      2answers
      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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      1answer
      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\...

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