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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.

        1
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        2answers
        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 ...
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        votes
        0answers
        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 ...
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        votes
        0answers
        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
        1answer
        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\...
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        votes
        0answers
        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 ...
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        votes
        0answers
        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
        votes
        0answers
        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 ...
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        votes
        2answers
        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 ...
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        votes
        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
        votes
        2answers
        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
        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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        votes
        2answers
        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 $f{\restriction}C$ ...
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        votes
        1answer
        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
        votes
        0answers
        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-...
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        votes
        0answers
        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 $\mathbb R^3$ ...

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        山西福彩快乐十分钟
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