Q 64. - Calculus

Question

Prove that if S is a closed subset of $R^2$ or $R^3$, then $S^e$ is an open set. This is Theorem 12.12 

Step-by-Step Solution

Answer

It is proved that if S is a closed subset of $R^2$or $R^3$, then $S^e$ is an open set .

Step 1. Given information.

We have given closed subset $S$ of $R^2$ or $R^3$.

Step 2: Prove the given statement.

The objective is to prove the theorem which states that, "If S is a closed subset of $R^2 or R^3$ then  $S^e$is an open set."

 Assume S is a closed subset of $R^2 or R^3$.

The definition of a closed set is given as, “A subset of $R^2$ or $R^3$is said to be closed, if its complement is an open set."

Thus, from the above definition, S can be termed to be closed, if its complement $S^e$ is open. Combining the two statements, it is proved that if S is a closed subset of $R^2$or $R^3$, then $S^e$is an open set .