Is discrete metric space connected?

A metric space X is connected if, and only if, its only connected component is X. In a discrete metric space, every singleton set is both open and closed and so has no proper superset that is connected. Therefore discrete metric spaces have the property that their connected components are their singleton subsets.

Which is separable space?

A topological space (X,τ) is said to be a separable space if it has a countable dense subset in X; i.e., A⊆X, ¯A=X, or A∪U≠ϕ, where U is an open set. In other words, a space X is said to be a separable space if there is a subset A of X such that (1) A is countable (2) ¯A=X (A is dense inX).

Is discrete metric space closed?

As any union of open sets is open, any subset in X is open. Now for every subset A of X, Ac = X\A is a subset of X and thus Ac is a open set in X. This implies that A is a closed set. Thus every subset in a discrete metric space is closed as well as open.

Is metric function continuous?

A function from one metric space to another, f:A→B, is continuous at p if for all ϵ>0 there exists δ>0 such that d(x,p)<δimpliesd(f(x),f(p))<ϵ. If f is continuous at all p∈A then we say that f is continuous on A or simply continuous.

Is the discrete metric closed?

Is R with discrete topology separable?

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.

Why is it called separable space?

Maybe, it refers to the separation of points by means of countable collection of small open sets (where separation means that for any two points we may find their disjoint neighborhoods in our collection).

What’s the meaning of separable?

capable of being separated or dissociated
Definition of separable 1 : capable of being separated or dissociated separable parts. 2 obsolete : causing separation.

Is discrete metric space open?