Q 65. - Calculus

Question

Let S be a subset of $R^2$ or $R^3$. Prove that a set S is open if and only if $\partial S\cap S=\varnothing$

Step-by-Step Solution

Answer

It is proved that a set S is open if and only if $\partial S\cap S=\varnothing$.

Step 1. Given information.

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

Step 2: Prove the given statement.

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

Assume an element x' such that $x\in S$.where S is an open set. 

A set is said to be open if for every element of it, there exists an open disk or ball D, such that 

$x\in D\subseteq S$

This would mean that 'x' does not belong to the boundary of S .

$x\notin ds$

Thus, there is no common element between $S$ and $\partial S$ is $\partial S\cap S=\varnothing$.

In another case, consider S is not an open set. 

Thus, the set $D\cap S^e$ or $\partial S\cap S$ is non-empty.

 Combining the two proofs, it is proved that, "that the set is open if and only if $\partial S\cap S=\varnothing$"