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

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        Connected and homogeneous $T_2$-space not homeomorphic to a subset of $\mathbb{R}^n$

        What is an example of an connected and homogeneous $T_2$-space $(X,\tau)$ with $|X|=2^{\aleph_0}$ such that for no $n\in\mathbb{R}$ the space $(X,\tau)$ is homeomorphic to a subspace of $\mathbb{R}^n$?...
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        How to prove that the set of the diffeomorphisms is open set in the space $C^1$ [on hold]

        How to prove that the set of the diffeomorphisms is open set in the space $C^1$? Can anyone indicate a book, or give some suggestion?
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        87 views

        What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

        In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales: ...the single most important fact which distinguishes locales from spaces: the ...
        4
        votes
        1answer
        113 views

        Is the Euclidean topology on $\mathbb{R}$ contained in a maximal connected topology?

        If $(X,\tau)$ is a connected space, then $\tau$ need not be contained in a maximal connected topology. Is the Euclidean topology on $\mathbb{R}$ contained in a maximal connected topology?
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        88 views

        CW-complex structure on the quotient

        Let $X$ be an $n$-dimensional CW-complex and let $A \subseteq X$ be a subcomplex. I want to show that the quotient space $X/A$ admits a structure of a CW-complex with skeletons $(X/A)^j := \pi(X^j \...
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        vote
        1answer
        82 views

        The preimage of continuum on Torus

        Let $p:\mathbb{R}^2\rightarrow\mathbb{R}^2/\mathbb{Z}^2$ be the natural projection, obviously $\mathbb{R}^2/\mathbb{Z}^2$ is the torus $\mathbb{T}^2$, if $K$ is a connected and compact subset of $\...
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        votes
        1answer
        91 views

        $T_2$-spaces in which no two open sets are homeomorphic

        This question was about spaces in which all non-empty open sets "look alike". Now I am interested in the opposite: Is there a $T_2$-space $(X,\tau)$ with $|X|>1$ such that whenever $U\neq V$ are ...
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        1answer
        58 views

        Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

        Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...
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        votes
        1answer
        129 views

        effectively distinguishing knots

        It was proven, I think by Mijatovi? EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound,...
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        63 views

        A community effort: equilibrium in quitting games

        This thread is in the spirit of the polymath project: a combined effort of the community to solve a difficult open problem. It is an activity of the European Network for Game Theory whose goal is to ...
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        1answer
        41 views

        Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

        This is a cross-post to the question I asked at MSE. The set of Schur stable matrices is \begin{align*} \mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\}, \end{align*} where $\rho(\cdot)$ denotes ...
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        52 views

        Topological Shape Operator More Sensitive than Inverse Limits

        This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...
        3
        votes
        1answer
        62 views

        sequences of iterated orthogonals (lifting property) in a category

        I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property. For example, several iterated orthogonals of $ \emptyset\...
        2
        votes
        1answer
        72 views

        Is the topology generated by the union of a chain of paracompact topologies paracompact?

        Let $X$ be a set and let ${\frak T}$ be a collection of paracompact topologies on $X$ such that for any $\tau, \tau'\in {\frak T}$ we have $\tau\subseteq \tau'$ or $\tau'\subseteq \tau$. Let $\sigma$ ...
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        vote
        1answer
        149 views

        Is every paracompact topology contained in a maximal paracompact topology?

        If $(X,\tau)$ is a paracompact, is there a topology $\tau'\supseteq \tau$ such that $(X,\tau')$ is still paracompact, and $\tau'$ is maximal with respect to $\subseteq$ and paracompactness?

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        山西福彩快乐十分钟
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