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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.

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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).

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  • $\begingroup$ 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
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    $\begingroup$ Yes. If the space is metrizable, you can use any property of metric spaces. $\endgroup$ – Tom Mar 18 '15 at 6:41

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