# Tagged Questions

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

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### Topological properties via properties continuous maps

A topological space $(X,\tau)$ is connected if and only if the only continuous maps $f:X\to\{0,1\}$ (where $\{0,1\}$ carries the discrete topology) are the constant maps. Are there other examples of ...
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### Does the topology induced by the Hausdorff-metric and the quotient topology coincide?

Assume that $X$ is a metric space, and $\sim$ is an equivalence relation on $X$. Furthermore we assume that the number of elements in each equivalence class is bounded by a positive constant. Does ...
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### Which spaces have enough curves

Let $\mathbf{Top}$ be the category of topological spaces, and let $I\in\mathbf{Top}$ be the unit interval $I=[0,1]\subset\mathbb{R}$. For any space $X$, let $|X|$ denote the underlying set of points; ...
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### Set of w*-continuous operators closed for the weak* topology or not?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
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### What is a generalized limit?

In the proof of Lemma 1.3 in the paper "The ideal structure of a groupoid C* algebra", Journal of Operator Theory 1991 by Jean Renault, I found the notion of a generalized limit of a net without any ...
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### Neighborhoods with proper multiplication

The following question was originally asked here, by C. Dubussy: http://math.stackexchange.com/questions/1802111/neighbourhoods-with-proper-multiplication Assume we have two closed subsets $F$ and $G$...
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### Topology on the space of Borel measures

Let $B$ be the set of all measures $\phi$ of $\mathbf{R}^{n}$ such that every open set is $\phi$-measurable (sometimes these measures are called Borel measures). Note the measures in $B$ are ...
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### An example of a regular but not well-based topological space

Call a topological space $\langle X,\mathscr{O}\rangle$ regular iff it is both $T_0$ and $T_3$: for every point $x\notin A$, where $A$ is a closed subsets of $X$, there are open and disjoint sets $V$ ...
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### Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15): Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the Gromov-...
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