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Chapter 12: Q 68. (page 932)

Let S be a subset of R2orR3. Prove that ∂S is a closed set.

Short Answer

Expert verified

It is proved that if S is the subset then∂Sis the closed set.

Step by step solution

01

Step 1. Given information. 

We have givenS is a subset ofR2orR3.

02

 Prove the given statement. 

Assume an element 'x'such thatx∈∂S

This means that the element 'x'is on the boundary ofS.
Every open disk D around this element x' will intersect bothS and Sc.
The complement of ∂S will be union of Sand Sc, excluding the set ∂Sitself.
If S is an open set, then Scis a closed set, and vice versa.
This can be possible only if the complement of ∂Sis an open set.
This is the definition of closed set.
Thus, it is proved that ∂Sis a closed set.

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