In this video, we review some basic concepts of metric space and then prove that every metric space is a topological space therefore a metric space turns out to be a crucial and beautiful example of a topological space. We also indicate that not every topological space is a metric space. Furthermore, We provide a definition of the metrizability of a topological space.
Timestamps:
0:00 - Introduction
1:07 - Definition of Metric Space
6:22 - Example: Discrete Metric Space
8:51 - Definition of open ball in Metric Spaces
10:14 - Definition of open set in Metric Spaces
12:31 - Example
14:41 - Every metric space is a topological space.
17:34 - Proof
35:31 - Definition of Metrizable spaces