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.