Activity 12.6.
Let
Hint.
Assume that such a metric exists and consider the open balls centered at
Section Metric Spaces and Topological Spaces
Every metric space is a topological space, where the topology is the collection of open sets defined by the metric. This topology is called the metric topology. A natural question to ask is whether every topological space is a metric space. That is, given a topological space, can we define a metric on the space so that the open sets are exactly the sets in the topology? For example, any space with the discrete topology is a metric space with the discrete metric.
We conclude that every metric space is a topological space, but not every topological space is a metric space. The topological spaces that can be realized as metric spaces are called metrizable.
