Questions tagged [metrizability]

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

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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 ...
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Topological space not being metrizable.

Question: Consider $\mathbb{R}^{2}$ with the topology $\mathcal{T} = \{U \subseteq \mathbb{R}^{2} : \mathbb{R}^{2} \setminus U \text{ is countable }\} \cup \{\emptyset\}$. Suppose that there is a ...
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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 ...
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If $E$ is the strict inductive limit of Frechet spaces $\lbrace E_k\rbrace$, why can't the countable basis of $E_1$ be appropriated for $E$?

Not wishing to dispute that strict LF-spaces are non-metrisable, I am trying to see the flaw in the intuitive sense I have that we could appropriate, for a strict LF space, a countable basis from the ...
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Locally compact Polish space admits a proper metric [duplicate]

If $X$ is locally compact Hausdorff, then the following are all equivalent: $X$ is second countable, $X$ is metrizable and $\sigma$-compact, $X$ is metrizable and separable, $X$ is Polish. I want to ...
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Coincidence of Subspace and Metric Topologies

How to show that a subspace of a metrizable space is also metrizable? Let $X$ be a topological space, and let $Y$ be a (non-empty) subset of $X$; let $d$ be a metric on $X$ that induces the topology ...
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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 ...
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Denumerable dimensional tvs

I am trying to prove the above. However, I even cannot understand why it is true. For example, $\ell^2$ is a complete normed space which has countable orthonormal basis and so countable dimensional. ...
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Why is Niemytzki plane locally metrizable?

The answer https://math.stackexchange.com/a/4411293/32337 includes the assertion that the Niemytzki plane $\Gamma$ (also known as the Moore plane, and as the tangent disk space) is locally metrizable. ...
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Measure Space without any form of metric

I'm with a difficult question that I was thinking . Any help will be useful We know that the Urysohn's Theorem states that: every Hausdorff second-countable regular space is metrizable. There are ...
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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 ?
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Showing that three statements are equivalent (Topology)

$(a)$ Assume that $X$ is a compact metrizable space. Prove that the following conditions are equivalent: $(i)$ $T$ is transitive. $(ii)$ The set of transitive points for $T$ is dense in $X$. $(iii)$ ...
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Perfect image of a metrizable space is metrizable.

How can I prove that the perfect image of a metrizable space is metrizable? I know the following three theorems about equivalent conditions of metrizability. A space $X$ is metrizable if and only if ...
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Show that $C(\mathbb R, \mathbb R)$ is not connected

$\bar{d}: \mathbb R × \mathbb R → \mathbb R$ denotes the standard bounded metric on $\mathbb R$ defined by $\bar{d}(x, y) := \min\{|y − x|, 1\}$. Given a topological space $X$, let $C(X, \mathbb R)$ ...
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Metrizability of the set of probability measures with the Kantorovich–Rubinstein metric

Let $X$ be a compact metric space and let $\mathcal{M}(X)$ be the set of probability measures. Then the weak topology on $\mathcal{M}(X)$ is metrizable, for example with the Wasserstein-metric and ...
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Prove on metrization of uncountable product [duplicate]

I am given the following problem: Show that an uncountable product of unit intervals is not first countable, and thus not metrizable. My answer would be that a), since the elements of the neighborhood ...
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Show strong topology on the closed unit ball is metrizable

Let $S$ be the closed unit ball of $B(H)$, the bounded operators on the Hilbert space $H$. I want to show that the relative strong topology on $S$ is metrizable. Attempt: I have already established ...
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Weak topology is not metrizable

I'm now reading some properties of weak topology, I have some problems which may related to the topology property in non-metrizable space ($E$ is a Banach Space): I know that $E^*$ with weak* ...
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Is it possible to show $\mathbb{R}^\omega$ is not metrizable without using the first countable definition and sequence lemma?

I encountered a question about proving that $\mathbb{R}^\omega$, which is the countably infinite product of $\mathbb{R}$ under the product topology, is not metrizable. I have seen many solutions here ...
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Metrization problem

Can the following problem be solved by Nagata Smirnov metrization theorem, which states that A topological space is metrizable if and only if it is frechet, regular and has a sigma locally finite ...
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Nagata-Smirnov Metrization Theorem for Pseudometric Spaces

The Nagata-Smirnov Metrization Theorem states that $X$ is metrizable iff it is $T_3$ and has a $\sigma$-locally finite base So, I was wondering if this holds for pseudometric spaces too, if we ...
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