0
$\begingroup$

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom.

I am having a hard time clearly understanding the difference between a Metric Topology and a Metrizable Topology, and how they relate with one another.

From what I understand:

  • A Metrizable Topology is a topological space that is generated by a metric space.
  • A topology induced by a metric defined on a metric space $X$ is called a Metric Topology.

Sadly it is still hard for me to distinguish the two topologies. I am curious as to whether or not Metrizable Topologies inherit topological properties from metric spaces?

The reason I ask this is because my professor asked I prove the following two implications:

  • A metrizable topology is first countable.
  • A separable and metrizable Topology is second countable.

I went off the assumption that metrizable topologies do indeed inheritable topological properties from metric spaces. Thus, I came up with a proof showing every metric space is first countable. I was also able to cooked up a proof showing that every separable metric space is second countable. If what I said about metrizable topologies is correct, then I want to say I am done with proving what the professor asked me to prove. However, I am not sure. Any suggestions?

I am still new to the study of topology so it is taking me some time to fully understand the material. Regardless, I sincerely thank you for taking the time to read this question. I greatly appreciate any assistance you may provide.

$\endgroup$
2
$\begingroup$

If the topology is metrizable, then there exists a metric that generates it, so it can be considered a metric topology corresponding to that metric. It then has all the properties of a metric space.

The reason for the different terminology is that sometimes you start with a metric and look at the topology you get (the metric topology), and sometimes you start with a topological space and ask whether there's some metric that induces that topology (i.e. whether the space is metrizable).

$\endgroup$
2
  • $\begingroup$ That being said, would my approach to prove what I desire be sufficient? I guess to be more clear on what I mean, would I need to start by stating something like this: "Since the space $X$ is metrizable, then all that we know about metric spaces will be true for $X$ by definition. (... rest of proof...)". $\endgroup$ – Kevin_H Mar 18 '15 at 6:13
  • 1
    $\begingroup$ Yes. If the space is metrizable, you can use any property of metric spaces. $\endgroup$ – Tom Mar 18 '15 at 6:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.