# 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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### Smooth functions with certain properties on the sphere

Let $f$ be a smooth function defined on the sphere such that the set of points where $f(x) - f(\tilde{x}_y)$ vanishes divides $\mathbb{S}^2$ into exactly four regions for all $y\in \mathbb{S}^2$, ...
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### Big list of Hochster dual concepts

Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
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### Hausdorff-Lipschitz continuity of cone correspondence

Let $\mathbb{R}_+$ denote the strictly positive real numbers, let $\mathcal{X} \subset \mathbb{R}^n$ and $\mathcal{P} \subset \mathbb{R}^m$ be compact and convex subsets, let f: \...
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### Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold intersection is empty. Then is some $n$-fold intersection empty?

Cover the $n$-disc with $n+1$ open sets $D^n = U_0 \cup \dotsb \cup U_n$. Suppose that $U_0 \cap \dotsb \cap U_n = \emptyset$. Suppose moreover that the cover is irredundant in the sense that no ...
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### Relationship between quotient CW-complexes after attaching cells

I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
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### Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
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### Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
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Prove the inequality $$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$ where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\... 5 votes 1 answer 259 views ### Quotients in categories of metric spaces There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ... • 15.5k 0 votes 0 answers 142 views ### Connectedness of deleted symmetric product Let$X$be a connected Hausdorff space. It is well-known that the$n$-fold symmetric product$\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$is a connected space equipped with the Vietoris ... • 622 2 votes 0 answers 387 views ### $$\left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed] If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(... • 625 8 votes 1 answer 409 views ### The cardinality of projections of subsets of the Hilbert cube by inner products I have three related questions. Question 1: Is there a subset X of the Hilbert cube [0,1]^{\Bbb N} of cardinality continuum, such that for each sequence a\in [0,1]^{\Bbb N} with \sum a_n ... • 3,104 6 votes 1 answer 139 views ### Is there a Bernstein subset X of \mathbb{R} such that no continuous map f : X → [0,1] is surjective? Is there a Bernstein subset X of \mathbb{R} such that no continuous map f : X → [0,1] is surjective ? • 1,071 2 votes 0 answers 24 views ### Continuity of Kernel Mean Embeddings Given some kernel k: X \times X \to \mathbb{R} with RKHS H_k we say that k is characteristic on the space of signed Radon measures over X, denoted by \mathcal{M}(X), if the kernel mean ... • 81 1 vote 0 answers 62 views ### "Star" of a CW-complex Suppose we have a CW-complex X with a 0-cell e^0. Is the union of all the cells (of higher dimensions) for which e^0 is a boundary point open in X? I don't know if it has a name, but a similar ... • 11 0 votes 1 answer 48 views ### Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K? [closed] Let S be a compact connected surface. Let K be a compact subset of S and suppose that K has a finite number of connected components. Let U be a connected component of S \setminus K and ... 2 votes 1 answer 161 views ### Factorization systems for vector bundles Are there any well-known factorization systems for the category of vector bundles defined over topological spaces? • 47 1 vote 1 answer 110 views ### Variants of Dirichlet-type function as a pointwise limit of continuous functions Problem Suppose f is a function from a complete metric space X to a metric space Y, and suppose Y has points y_{0}, y_{1} such that the subsets f^{-1}(y_{0}) and f^{-1}(y_{1}) are both ... • 129 3 votes 1 answer 171 views ### Is there a metric separable space with the following properties...? Let \omega_1<\mathfrak{q}_0 where \mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}, Y is not a Q-space\}. Is there a metric separable space X with the following properties: |X|\geq\... • 1,071 0 votes 0 answers 82 views ### Finding an example if it exists, for a non-contractible and contractible space with special requirement on quotients of their union? Let A and B be subsets of n-dimensional Euclidean space \mathbb{R}^{n}, such that A is non-contractible, B is contractible and B is not an one-point set. I would like to find example(s) ... 0 votes 0 answers 49 views ### Approximating open subset of profinite group by union of cosets of ideal I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset U_P \subseteq \hat{\mathcal{O}}_P can be ... 3 votes 0 answers 93 views ### Finitely generated Banach lattice C(K) and partitions of unity Let E be a Banach lattice. A Banach sublattice L of E is called finitely generated if there exists a finite subset F \subseteq E such that$$L = \bigcap \{ \hat{L} \mid F \subseteq \hat L, \, \... • 419 5 votes 0 answers 79 views ### When a compact subset of a TVS can be continuously projected on a closed linear subspace? Let$V$be a (Hausdorff) topological vector space,$W\subset V$a closed linear subspace,$X\subset V $a compact. (Q): When there is a continuous map$P:X\to W$such that$P(x)=x$for every$x\in X\...
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Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume ...