How do you prove a metric space is separable?

How do you prove a metric space is separable?

We say a metric space is separable if it has a countable dense subset. Using the fact that any point in the closure of a set is the limit of a sequence in that set (yes?) it is easy to show that Q is dense in R, and so R is separable. A discrete metric space is separable if and only if it is countable.

Is D XY )=| xy 2 a metric space?

The answer is yes, and the theory is called the theory of metric spaces. A metric space is just a set X equipped with a function d of two variables which measures the distance between points: d(x, y) is the distance between two points x and y in X.

Is every metric space is separable?

Abstract. We first show that in the function realizability topos RT(K2) every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every T0-space is separable and every discrete space is countable.

How do you show that two metrics are equivalent?

Two metrics are equivalent if they give rise to the same topology. So you need to show that if a set is open in the topology induced by the metric, then it is also open in the topology induced by the metric, and vice-versa.

Is there relation between separable space and compact space?

We also have the following easy fact: Proposition 2.3 Every totally bounded metric space (and in particular every compact met- ric space) is separable. Intuitively, a separable space is one that is “well approximated by a countable subset”, while a compact space is one that is “well approximated by a finite subset”.

How do you find metric space?

A metric space is a set X together with such a metric. The prototype: The set of real numbers R with the metric d(x, y) = |x – y|. This is what is called the usual metric on R. The complex numbers C with the metric d(z, w) = |z – w|.

Is separable metric space Compact?

What is meant by topologically equivalent?

Definition of topological equivalence : the relationship of two geometric figures capable of being transformed one into the other by a one-to-one transformation continuous in both directions.