# Questions tagged [metrizability]

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

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### $(\prod_{i\in I}X_i, \mathcal{T}_p)$ is metrizable iff $X_i$ is metrizable and all $X_i$ is singleton set except countable many indices.

Note that $X_i$ is nonempty. I guess that a singleton space as a factor of the product topology doesn't affect the whole product. But I don't know how to show that in detial. Just some hints are ...
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### Show that if $X$ is compact metrizable then $C(X)$ is separable.

Let $X$ be a metrizable compact space. I want to show that $C(X, \mathbb{C})$ is separable in the uniform topology. Attempt: By our assumption, $X$ is separable, so we can pick a countable dense ...
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### Prob. 6, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: $\mathbb{R}_\ell$ and $I_0^2$ are not metrizable

Here is Prob. 6, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that $\mathbb{R}_\ell$ and $I_0^2$ are not metrizable. My Attempt: First, we consider $\mathbb{R}_\ell$. ...
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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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### 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 ...
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'\}....