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Chapter 12: Q 65. (page 932)

Let S be a subset of R2or R3. Prove that a set S is open if and only if 鈭�S鈭�S=鈭�

Short Answer

Expert verified

It is proved that a set S is open if and only if鈭�S鈭�S=鈭�.

Step by step solution

01

Step 1. Given information. 

We have given S be a subset of R2orR3.

02

 Prove the given statement. 

We have given S be a subset of R2orR3.

Assume an elementx' such that x鈭�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鈭�D鈯�S

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

x鈭�ds

Thus, there is no common element between Sand 鈭�Sis 鈭�S鈭�S=鈭�.

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

Thus, the set D鈭�Seor 鈭�S鈭�Sis non-empty.

Combining the two proofs, it is proved that, "that the set is open if and only if鈭�S鈭�S=鈭�"

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