Metric Spaces and Topological Spaces

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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.

Activity 12.6.

Let \(X = \{a,b,c,e\}\) and \(\tau = \{\emptyset, \{a\}, \{b\}, \{a,b\}, X \}\text{.}\) Explain why there cannot be a metric \(d : X \times X \to \R\) so that the open sets in the metric topology are the sets in \(\tau\text{.}\)

Assume that such a metric exists and consider the open balls centered at \(c\text{.}\)

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.

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