Questions tagged [metrizability]

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

Filter by
Sorted by
Tagged with
2
votes
1answer
41 views

$(\prod_{i\in I}X_i, \mathcal{T}_p)$ 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 ...
3
votes
1answer
47 views

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 ...
0
votes
1answer
34 views

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 $\mathbb{R}_\ell$. ...
13
votes
3answers
866 views

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 $\...
1
vote
2answers
45 views

Is the topology on compact connected Lie groups metrizable?

The question is in the title, really. Suppose $G$ is a compact connected Lie group. Is there a metric on $G$ which induces the underlying topology? (so in particular $G$ is compact and connected wrt ...
0
votes
1answer
45 views

In a totally ordered topological space that is connected and metrizable, does every sequence have a monotone subsequence?

I am wondering if in a totally ordered topological space (order topology) that is connected (hence linear continuum) and metrizable, does every sequence have a monotone subsequence? Can we generalize ...
0
votes
1answer
46 views

Monotone convergence theorem for sequences in a completely metrizable totally ordered topological space

In the Reals it is commonly known that a monotone bounded sequence is convergent. This is due to the first countability axiom of the metric space and the least upper bound property of the reals. Can ...
0
votes
1answer
80 views

Is a totally ordered, separable and connected topological space metrizable (in the order topology)?

Is a totally ordered, separable and connected topological space metrizable (in the order topology)? If we relax the assumption of connectedness, I know the counterexamples, but if we have a linear ...
1
vote
4answers
215 views

Metric spaces are metrizable?

I know a topological space need not be a metric space and every metric space can be considered a topological space (which is the one induced by a metric defined on it). But, I've come across this ...
0
votes
1answer
50 views

What qualifies as a “separation axiom?”

Wikipedia states the hierarchy of separation axioms as: $$ \underset{\text{(Kolmogorov)}}{T_0} \impliedby \underset{\text{(Fréchet)}}{T_1} \impliedby \underset{\text{(Hausdorff)}}{T_2} \impliedby \...
2
votes
1answer
82 views

Nagata-Smirnov vs. Urysohn metrization theorems - an example?

I'm looking for an example to demonstrate that fact that the Nagata-Smirnov thm is "more useful" than Urysohn's. That is, I'm looking for a space that you can prove is metrizable using Nagata-Smirnov ...
1
vote
3answers
144 views

Countable product of metric spaces is metrizable [General Metric]

I know that if we have a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$ then $X=\Pi^{\infty}_{n=1}X_n$ is a metric space with metric $\rho((x_n)_{n \in \mathbb{N}},(y_n)_{n \...
5
votes
2answers
74 views

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 $x \in X - C$...
1
vote
1answer
47 views

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 ...
0
votes
1answer
20 views

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 ...
1
vote
1answer
58 views

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 : ...
1
vote
1answer
70 views

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 ...
0
votes
1answer
48 views

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 ...
1
vote
1answer
61 views

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\...
4
votes
2answers
94 views

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 ...
0
votes
1answer
43 views

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
1
vote
0answers
86 views

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 ...
5
votes
1answer
186 views

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 ...
2
votes
1answer
300 views

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. " ...
1
vote
0answers
135 views

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 $E$...
3
votes
1answer
446 views

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 ...
-4
votes
1answer
149 views

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'\}....
3
votes
3answers
2k views

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 $...
2
votes
1answer
220 views

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 ...
1
vote
1answer
542 views

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 ...