How to prove a topological space is metrizable

In summary, X is a set and P(X) is the discrete topology on X. To prove that X is metrizable, we need to find a metric that induces the discrete topology. This can be done by using the discrete metric, which is defined by d(x,y) = 0 if x=y and d(x,y) = 1 if x=/=y. From this, we can show that every subset of X is open and therefore, (X, P(X)) is a topological space. This also proves that (X, d) is a metric space. Hence, X is metrizable.
  • #1
variety
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Homework Statement


X is a set and P(X) is the discrete topology on X, meaning that P(X) consists of all subsets of X. I want to prove that X is metrizable.

Homework Equations


My text says that a topological space X is metrizable if it arises from a metric space. This seems a little unclear to me, which is probably why I am slightly confused.

The Attempt at a Solution


This problem seems really easy, I am just unsure of what I am supposed to prove. I want to show that X, together with the discrete topology, is metrizable. I choose the discrete metric d, which is defined by d(x,y) = 0 if x=y and d(x,y) = 1 if x=/=y.

This is where I am unsure of what I am supposed to show. I can start by showing that if we have the metric space (X,d), then every subset of X is open since all the points are isolated. Then P(X) satisfies the property of a topology on X, so (X,P(X)) is a topological space. But I don't think this is correct because we already assumed (X,P(X)) is a topological space. In fact, it is ALWAYS a topological space for any X, right?

Am I supposed to show that if (X,P(X)) is a topological space, (X,d) is a metric space? But this is also obvious because I already know that d is a metric.

What am I supposed to be proving?
 
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  • #2
You start with a space with the discrete topology. You want to find a metric that induces the discrete topology. You already did that with the discrete metric (did you show it's a metric?) and you showed it induced the discrete topology which you did when you said "I can start by showing that if we have the metric space (X,d), then every subset of X is open since all the points are isolated." I'll admit it's not a hard proof, but it's not completely without substance.
 
  • #3
counter example of topological space which is not metric space
 
  • #4
variety said:
But I don't think this is correct because we already assumed (X,P(X)) is a topological space. In fact, it is ALWAYS a topological space for any X, right?

Am I supposed to show that if (X,P(X)) is a topological space, (X,d) is a metric space? But this is also obvious because I already know that d is a metric.

What am I supposed to be proving?

Given any set X, (X, P(X)) is a topological space, this is a fact, and your starting assumption.

Of course you know that d is a metric. You found a metric which induces the discrete topology. As Dick noted, there's not much more to it.
 

1. Can every topological space be equipped with a metric?

No, not every topological space can be equipped with a metric. A topological space is metrizable if and only if it is Hausdorff and second countable. Therefore, if a topological space does not satisfy these conditions, it cannot be equipped with a metric.

2. What is the definition of a metrizable topological space?

A topological space is metrizable if it can be equipped with a metric that induces the same topology as the original topological space. In other words, every open set in the topological space can be written as a union of open balls in the metric space.

3. How do you prove that a topological space is metrizable?

To prove that a topological space is metrizable, you need to show that it satisfies the necessary conditions of being Hausdorff and second countable. Then, you need to construct a metric that induces the same topology as the given topological space. This can be done by defining a distance function between points in the topological space and showing that it satisfies the properties of a metric.

4. Can a topological space have more than one metric that induces the same topology?

Yes, a topological space can have more than one metric that induces the same topology. This is because a metric space can have multiple distance functions that satisfy the properties of a metric. However, the resulting metric spaces will still be homeomorphic to each other and therefore, they will induce the same topology on the original topological space.

5. Are all metric spaces also topological spaces?

Yes, all metric spaces are also topological spaces. This is because the metric on a metric space defines a topology on the set of points, where the open sets are defined as unions of open balls in the metric space. However, not all topological spaces can be equipped with a metric, as mentioned in the first question.

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