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Questions tagged [metrizability]

For questions pertaining to the metrizability of topological spaces and / or metrization theorems.

We call a topology on a set $X$ metrizable when we can endow $X$ with a metric that induces this topology. Determining whether a space is metrizable is, historically, a major problem in point-set topology, and much work has been dedicated to finding relevant necessary and/or sufficient conditions with results such as Urysohn's metrization theorem, the Nagata–Smirnov metrization theorem, et al.

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A topology that is finer than a metrizable topology is also metrizable?

If $\tau_{1}$ and $\tau_{2}$ are two topologies on a set $\Omega$ such that $\tau_{1}$ is weaker than $\tau_{2}$ (i.e. $\tau_{1}\subset\tau_{2}$) and $\tau_{1}$ is metrizable, is it then true that $\tau_{2}$ is also metrizable? My guess is that it…
Calculix
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Is "ternary metrizability" equivalent to pseudometrizability?

Below, $X$ is always a set with at least three elements to avoid triviality. Say that a ternary metric on a set $X$ is a map $t:X^3\rightarrow\mathbb{R}$ with the following properties: Non-negativity: $t(x_1,x_2,x_3)\ge 0$ and $t(x_1,x_1,x_2)=0$.…
Noah Schweber
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The $\mathbf{F}$-metric induces the weak topology on the set of bounded varifolds

Some preliminary definitions and notation: (1) Given a vector space $\mathbb{V}$, we denote by $G_k(\mathbb{V})$ the $k$-grassmannian of $\mathbb{V}$, i.e. the set of all $k$-dimensional vector subspaces of $\mathbb{V}$; (2) Given a differential…
Derso
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A metrizable space is realcompact iff it has non-measurable cardinality?

A space is realcompact if its a closed subspace of an arbitrary product of real lines, with product topology. A cardinal $\kappa$ is called measurable if there exists a (countably additive) $\{0, 1\}$-valued measure $\mu:\kappa\to \{0, 1\}$ with…
Jakobian
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How did submetrizability come into existence ? What is submetrizability of a topological space used for?

Can anybody tell me how did submetrizability come into existence, and what is its use in topology ? Any examples to make me understand ?
Mir Aaliya
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Metrizability is a topological property?

How could I show that metrizability is a topological property? Well, this means that if I have a set $X$ that is metrizable and a homeomorphic function $f$ from $X$ to $Y$, then I need to show that $Y$ is metrizabke, correct? If I let $d$ be a…
Akaichan
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Separability and the Nagata-Smirnov Metrisation Theorem

Definitions: Let $X$ denote a topological space throughout. If all singleton subsets of $X$ are closed, then we call $X$ Fréchet. If, given any closed subset $C \subset X$ and any point $x \in X - C$, there exist disjoint neighbourhoods of $x$ and…
Jack
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A Cech-complete subspace is a $G_\delta$ in its closure

How does one prove the following result (Engelking, exercise 3.9.A): If a Cech-complete space $X$ is a subspace of a Hausdorff space $Y$, then there exists a $G_\delta$ set $Z \subseteq Y$ such that $X=\overline{X}\cap Z$. In other words, if $X$…
PatrickR
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$L^p_{loc}(\Omega)$ is completely metrizable

Let $\Omega \subset \mathbb{R}^n$ be a (not necessarily bounded) domain and $1 \leq p \leq \infty$. Then define $L^p_{loc}(\Omega)$ to be the set of functions $f: \Omega \rightarrow \mathbb{R}$ such that $$\Big(\int_K |f|^p\Big)^{1/p} < \infty,…
CBBAM
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Special Type of Locally Metrizable Space

We say a topological space is locally metrizable if for every $x\in X,$ there is an open set $U$ containing $x$ which is metrizable. That is, the subspace topology on $U$ is the topology induced by some metric $d_U.$ In my research, I've come across…
Miles Gould
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Is the difference between metric spaces and ultrametric spaces topological?

My question is exactly what is written in the title. A metric space is a set $X$ equipped with a function $d: X \times X \to \mathbb{R}_{\geq 0}$ satisfying $d(x, y) = 0 \iff x = y$, $d(x, y) = d(y, x)$, and the triangle inequality: $d(x, y) \leq…
Joe Stephen
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Can a topological embedding of a metric space into a metrizable space be extended to an isometric embedding for some metric on the codomain?

Motivation: Consider for example a metric space that is a disjoint union of a point with $\mathbb R$ (with the usual metric on $\mathbb R$). Intuitively, it feels like there is some space "missing" from it. Topologically we can fix that by embedding…
Carla_
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Construct a metric that induces a given topology

In a topology textbook there was a exercise to determine the topology induced by $$x^2:\mathbb{R}\to\mathbb{R}$$ where the target has the euclidean topology. I am the opinion that $x^2$ induced a kind of "mirrored" topology, meaning the open sets…
HyperPro
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Is the "unit sphere" in $\mathbb{R}^\omega$ metrizable?

Let $\mathbb{R}^\omega$ be the countably infinite product of $\mathbb{R}$ with itself in the product topology. $\mathbb{R}^\omega$ is metrizable but the metric doesn't arise from a norm, so a natural analogue to a unit sphere is the quotient of…
Compact_Ball
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How to prove that the following metric induces the subspace topology?

I am trying to follow Theorem (3.11) of Kechris's Classical Descriptive Set Theory. In this part of the proof he shows that a $G_{\delta}$-subspace Y of a completely metrizable space $(X,d)$ is completely metrizable. For this, he defines the…
a_hayler
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