Q 68. - Calculus

Question

Let S be a subset of $R^2 or R^3$. Prove that $\partial S$ is a closed set. 

Step-by-Step Solution

Answer

It is proved that if S is the subset then $\partial S$ is the closed set.

Step 1. Given information.

We have given S is a subset of $R^{2 }or R^3$.

Step 2: Prove the given statement.

Assume an element 'x'such that $x\in \partial S$ 

This means that the element 'x' is on the boundary of S. 

Every open disk D around this element x' will intersect both S and $S^c$ .

The complement of $\partial S$ will be union of S and $S^c$, excluding the set $\partial S$ itself.

 If S is an open set, then $S^c$ is a closed set, and vice versa.

This can be possible only if the complement of $\partial S$ is an open set. 

This is the definition of closed set. 

Thus, it is proved that $\partial S$ is a closed set.