Q 70. - Calculus

Question

Prove that the empty set is both an open subset and a closed subset of $R^2$. 

Step-by-Step Solution

Answer

It is proved that the empty set is both an open subset and a closed subset  of $R^2$.

Step 1. Given information.

We have given subset of $R^2$.

Step 2: Prove the given statement.

The objective is to prove that the given set is both open and closed. 

A subset is said to be open if it does not have a boundary to identify.

Consider an empty set. It does not have any boundary to identify. So, it is an open set. The complement of an empty set is the universal set, which again has no boundary. So, universal set is also an open set. Hence, its complement, an empty set, must be a closed set.
 Thus, an empty set $\varnothing \subset R^2$  is Both Open and Closed Set.