Metric Spaces: Metrizability and Homeomorphisms

If a topological space \( X \) is metrizable and \( Y \) is homeomorphic to \( X \), then \( Y \) is also metrizable.

In other words, metrizability is a property that remains unchanged under homeomorphisms.

This means that if a space \( X \) is metrizable, then any space that is homeomorphic to \( X \) must also be metrizable.

So, if I already know that a space \( X \) is metrizable and I find another space \( Y \) that is homeomorphic to \( X \), I don’t need to explicitly construct a metric for \( Y \). I can immediately conclude that \( Y \) is metrizable as well.

Explanation

A topological space \( X \) is metrizable if there exists a metric \( d \) that induces its topology. In other words, the topology of \( X \) can be fully described in terms of a metric.

A homeomorphism is a bijective function between two topological spaces \( X \) and \( Y \) that is continuous and has a continuous inverse. If \( X \) and \( Y \) are homeomorphic, they share the same topological properties because a homeomorphism preserves the underlying topological structure.

Now, if \( X \) has a metric \( d \) that induces its topology, then \( Y \), being homeomorphic to \( X \), inherits a topological structure that can also be described by a metric.

Put simply, if a metric exists for \( X \), then a corresponding metric can be defined for \( Y \) as well, since \( X \) and \( Y \) are topologically equivalent.

A Practical Example

Consider the real line \( \mathbb{R} \) with its standard topology induced by the Euclidean distance, and the open interval \( (-1, 1) \).

The space \( \mathbb{R} \) is metrizable with the Euclidean metric.

Now, define the function \( f : \mathbb{R} \to (-1,1) \) by \( f(x) = \frac{x}{\sqrt{1+x^2}} \). This function is a homeomorphism because it continuously "compresses" the entire real line into the open interval \( (-1,1) \), while remaining bijective, and its inverse is also continuous.

Since \( f \) is a homeomorphism, it preserves the topological structure.

Therefore, because \( \mathbb{R} \) is metrizable, the homeomorphic space \( (-1,1) \) must also be metrizable—and indeed, it is, with the Euclidean metric.

And so on.