Sage-Tips

Is a Metrizable space a metric space?

There is no difference between a metrizable space and a metric space (proof included).

What is difference between metric and metric space?

A metric space is a set where a notion of distance (called a metric) between elements of the set is defined. A normed space is a vector space with a special type of metric and thus is also a metric space.

Which spaces are metrizable?

Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and hence normal and Tychonoff) and first-countable. However, some properties of the metric, such as completeness, cannot be said to be inherited.

Under what conditions does a metrizable space have a metrizable compactification?

Under what conditions does a metrizable space have a metrizable compactification? SOLUTION. If A is a dense subset of a compact metric space, then A must be second countable because a compact metric space is second countable and a subspace of a second countable space is also second countable.

Why every topological space is not metric space?

Not every topological space is a metric space. However, every metric space is a topological space with the topology being all the open sets of the metric space. That is because the union of an arbitrary collection of open sets in a metric space is open, and trivially, the empty set and the space are both open.

Which product of two metric spaces is a metric space?

Products of two metric spaces: The product of two metric spaces (Y,dY ) and (Z, dZ) is the metric space (Y × Z, dY ×Z), where dY ×Z is defined by dY ×Z((y, z),(y ,z )) = dY (y, y ) + dZ(z,z ).

Why every topological space is not a metric space?

Under what conditions does a Metrizable space have a Metrizable compactification?

Is a subspace of a metrizable space metrizable?

A subspace of a completely metrizable space X is completely metrizable if and only if it is Gδ in X. Hence, a product of nonempty metrizable spaces is completely metrizable if and only if at most countably many factors have more than one point and each factor is completely metrizable.