why does one want to metrize a topology?

Printable View

- Jun 27th 2006, 01:35 PMnweissmametrizing a topology
why does one want to metrize a topology?

- Jun 27th 2006, 07:04 PMThePerfectHacker
I was finally able to find this on PlanetMath, (Source taken from this website below).

A topological space $\displaystyle (X,\mathcal{T})$ is said to be metrizable if there is a metric $\displaystyle d:X^2\to \mathbb{R}^+$ such that the topology induced by $\displaystyle d$ is $\displaystyle \mathcal{T}$.

---

This is what bothers me,

*induced*as in graph theory.

By the problem is that the topology can contain more than two element subsets of $\displaystyle X$ in that case, how can we view $\displaystyle (X,\mathcal{T})$ as a graph!

Furthermore, $\displaystyle \{\} \in \mathcal{T}$ then for certainly the topology on $\displaystyle X$ does not contain two element subsets. That means that $\displaystyle G=(X,\mathcal{T})$ is not a graph and the entire concept of $\displaystyle \mathcal{T}= G[d]$ makes no sense.

---

Forgive, me for not answersing your question. It just bothered me too. - Jun 28th 2006, 12:05 AMJakeDQuote:

Originally Posted by**ThePerfectHacker**

Here's a definition of induced topology from*Topology*by James Dugundji.

Let Y be a set and d be a metric in Y. The topology T(d), having for basis the family { Bd(y,r) | y in Y, r > 0 } of all d-balls in Y, is called the topology in Y induced (or determined) by the metric d. - Jun 28th 2006, 01:27 AMJakeDQuote:

Originally Posted by**nweissma**

For example, the definition of continuity of a function that we learned in calculus

$\displaystyle f:X \to Y$ is continuous if $\displaystyle \langle x_n \rangle \to x$ implies $\displaystyle \langle f(x_n ) \rangle \to f(x)$

uses sequences. It applies when $\displaystyle X$ is a metric space but not a general topological space. But try teaching the general definition of continuity--the inverse image of an open set is open--to a first-year calculus student.

There are many theorems that apply to metric spaces but not general topological spaces. Thus it is important to know when a topology is metrizable so those theorems apply.

I have a book on probability measures on metric spaces. What matters there is that the topological spaces are metrizable, not the particular metric used.