What is a complete separable metric space?

What is a complete separable metric space?

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence. of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

Are metric spaces 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 every separable metric space is compact?

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”.

Is every Cauchy sequence convergent?

Every real Cauchy sequence is convergent. Theorem.

How do you know if metric space is complete?

A metric space (X, ϱ) is said to be complete if every Cauchy sequence (xn) in (X, ϱ) converges to a limit α ∈ X. There are incomplete metric spaces. If a metric space (X, ϱ) is not complete then it has Cauchy sequences that do not converge.

Is every metric space is second countable?

A space is first-countable if each point has a countable local base. Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.

Is real line separable?

Follows from: Real Number Line is Second-Countable. Second-Countable Space is Separable.

Is Lindelof space compact?

Properties of Lindelöf spaces In particular, every countable space is Lindelöf. A Lindelöf space is compact if and only if it is countably compact. Every second-countable space is Lindelöf, but not conversely. For example, there are many compact spaces that are not second countable.

What is a separable function?

Separable functions A function of 2 independent variables is said to be separable if it can be expressed as a product of 2 functions, each of them depending on only one variable.

Is Lindelof space Compact?

Can a Cauchy sequence diverge?

Each Cauchy sequence is bounded, so it can not happen that ‖xn‖→∞.