Question

Define what we mean by a separable space? Given an example of a separable space? Give an example of a non-separable space? Is there a relation between separable space and compact space? If yes, what is the relation? Justify your answer (prove)?

a) Separable space refers to a space with a dense countable subset.
b) Example of separable space: Euclidean space.
c) Example of non-separable space: Cantor set.
d) Yes, there is a relation between separable space and compact space, and it is proven in topology.

Answer 1

Separable space refers to a space with a dense countable subset. Examples include Euclidean space (separable) and the Cantor set (non-separable). There is a relation between separable space and compact space, where every compact space is separable, but not all separable spaces are compact.

Step-by-step explanation:

Learn more about Separable Space here:

Separable space refers to a space with a dense countable subset. For example, Euclidean space is a separable space because it contains a dense countable set, such as the rational numbers. On the other hand, the Cantor set is an example of a non-separable space because it does not contain a dense countable subset.

Yes, there is a relation between separable space and compact space. In fact, it is a proven result in topology. Every compact space is separable, but the converse is not necessarily true. In other words, all compact spaces are separable, but not all separable spaces are compact.