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Questions tagged [metrizability]

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

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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 $x \in X - C$...
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Covering characterization of Metrizability and Paracompactness

I am quite struck by the similarity between covering charachterization of Paracompactness and Metrizability both in the T1 and T3 topologies. In particular in the case of T1 topologies we have the ...
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In a complex metrizable topological vector space, $d(0,\alpha x)\neq |\alpha|d(0,x), \ \alpha \in \mathbb C.$

Let $(X,\tau)$ be a complex metrizable topological vector space with the metric $d$. Does the following hold: $$d(0,\alpha x)=d(0,x),\ \forall \alpha \in \mathbb C, |\alpha|=1 \ ?$$ In general, the ...
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Continuity of two variables in toplogical space .

Let $T$ is subset of real numbers .Let $k$ is any number which is not in $T$ . Define $$kT =\{ kt : t \in T\}. $$ Let $W$ is toplogy(not usual) on $T$ , then can you prove that : $$H=\{ kU : ...
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Does a continuous function preserve metrizability?

This is problem 6.1.8 of S. Morris's "Topology without Tears": Let $f$ be a continuous mapping of a metrizable space $(X,\tau)$ onto a topological space $(Y,\tau_1)$. Is $(Y, \tau_1)$ necessarily ...
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Metrizability of $\Bbb{R}^I$ and the Sequence Lemma

Let $I$ be some uncountable set, and let $\Bbb{R}^I$ denote the product of uncountably many copies of $\Bbb{R}$. Show that $\Bbb{R}^I$ is not metrizable. I know that there is a solution which ...
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To check whether given space is metrizable

Let $X$ be a compact Hausdorff topological space without isolated point . Also, there exist open cover $\mathcal{V}=\{V_i\}_{I=1}^m$ and a homeomorphism $f:X\to X$ such that $card(\bigcap _{j\in\...
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Urysohn Metrization Theroem : Proof of Injectivity

I am reading Munkres Topology and following given Urysohn metrization theorem and its proof, but can't understand why the injectivity simply given after define a index functions and product them. I ...
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Proof a theorem about Metrizable manifold

Where could the proof of the following theorem be found : a manifold is metrizable if and only if it is paracompact
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Proving a perfect map preserves metrizability

I'm trying to prove that a perfect map preserves metrizability using the Nagata-Smirnov Theorem, but I got stuck, hope someone can help me solve this. Nagata-Smirnov Metrization Theorem: A ...
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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 ...
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A space is metrisable if and only if it admits a countable basis

I have a topology question about metrisable and countable basis. Please indicate whether the argument below is true or not: "A space is metrizable if and only if it admits a countable basis. " ...
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metrizability of weak topology

I have a problem with a line at the end of an example who tolds this: let $(E,||-||)$ be a separable Banach space. If there exist a weak Cauchy sequence who doesn't converge weakly to an element of $E$...
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1answer
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Show that if a topological space is metrizable then it is so in an infinite number of ways

Show that if a topological space is metrizable then it is so in an infinite number of ways. Show that a topological space $X$ is metrizable $\iff$ there exists a homeomorphism of $X$ onto a subspace ...
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Metrizability of intersection of two topologies [closed]

Let $X$ be a space endowed with two topologies $\tau, \tau'$. Then one can show that $\tau\cap \tau'$ is also a topology on $X$, where $$ \tau \cap \tau' = \{ t \cap t' : t\in \tau, \ \ t'\in \tau'\}....
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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 $...
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1answer
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Doubts about metrization theorems

I studying metrization and in different parts I encountered different formulations of theorems, for example in the Nagata–Smirnov metrization theorem I found: A topological space $X$ is metrizable ...
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A condition for the image of a metrizable space to be metrizable

I noticed this problem on a previous exam that I completely missed, and I was wondering if anyone could help me out. Suppose $f: Y \rightarrow X$ is a continuous mapping of a separable metric space ...