# Tagged Questions

A metric space is a pair $(X,d)$, where $X$ is a set and $d:X \times X \to \mathbb{R}$ satisfies the following conditions for all $x,y,z \in X$. (Symmetry) $d(x,y)=d(y,x)$. (Identity of Indiscernibles) $d(x,y)=0$ if and only if $x=y$. (Triangle Inequality) $d(x,y)+d(y,z) \geq d(x,z)$.

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### BV functions with values in metric space

$\newcommand{\IR}{\mathbb{R}} \newcommand{\IN}{\mathbb{N}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\divergence}{\operatorname{div}} \newcommand{\Lip}{\operatorname{Lip}}$ Let $E$ be a ...
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### Choice and the Baire property in non-separable complete metric spaces

It's known to be consistent with ZF+DC that every subset of $\mathbb{R}$ has the Baire property (BP). (E.g. Shelah's model). If so, then every subset of every complete separable metric space has ...
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### Spaces that can't be embedded in the plane

If $X$ and $Y$ are topological spaces, let us write $X \preceq Y$ whenever $X$ embeds in $Y$. Earlier today, I asked the question: Is this a well-quasi-order on the completely metrizable spaces? ...
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### Conditions for a set being closed under taking complement of a ball twice

Given a subset $S$ of a finite metric space $F$ with a distance function $d(,)$ and a number $\delta > 0$ let $N_\delta(S) = \{x \in F| d(x,S)\ge \delta\}$. Is there a characterization of ...
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### Terminology for metrics?

For some reason, I'm currently interested in the following relation - let $d,\delta$ be two metrics on some space $X$. We call the metrics _______ if there are some constants $C,E>0$ such that for ...
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### A generalization of SOCA

Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset ...
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### Non-separable Banach space

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-...
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### Quotient of metric spaces

Let $(X,d)$ be a compact metric space and $\sim$ an equivalence relation on $X$ such that the quotient space $X/\sim$ is Hausdorff. It is well known that in this case the quotient is metrizable. My ...
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### Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
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### Nearly injective Banach spaces

There was a problem about nearly injective metric spaces posed by Aronszajn and Panitchpakdi which I actually solved in the past but it still remains open (as long as I know) for the Banach spaces--so ...
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### Continuity of central point operation

Stanisław Mazur and Stanisław Ulam, in their joint paper, characterized the mid-point $\ \frac{a+b}2\$ in a Banach space in pure metric terms (without algebra). This allowed them to show that any two ...
Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...