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Chapter 12: Q 66. (page 932)

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

Short Answer

Expert verified

It is proved that " the set is closed if and only if鈭�S鈯�S.

Step by step solution

01

Step 1. Given information.  

We have given S is a subset of R2or R3.

02

 Prove the given statement.  

The objective is to prove that the set is closed if and only if 鈭�S鈯�S

Assume an element x' such that X鈭�S, where Sis a closed set.
By the definition of a closed set, if Sis a closed set, then its complementSe is an open set.
This means that for element 'X' there exists such an open disk or ball D, such that D would include or extend beyond the boundary of S.
Thus, the set D鈭�Seor鈭�S鈭�S is non-empty.
From these statements, it is clear that 鈭�S鈯�S

In another case, consider 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 thatx鈭�D鈯�S
This would mean that 'x' does not belong to the boundary ofS.
x鈭�鈭�S
Thus, there is no common element between Sand 鈭�SThis means that as 鈭�S鈭�S
Combining the two proofs, it is proved that, "that the set is closed if and only if鈭�S鈯�S

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