Chapter 12: Q 67. (page 932)

Let S be a subset of $R^{2}$ or $R^{3}$ . Prove that $\partial S = \partial (S^{c})$

Short Answer

Expert verified

The given statement is proved if S is the subset then $\partial S = \partial (S^{c})$ .

Step by step solution

Step 1. Given information.

We have givenS is a subset of $R^{2} o r R^{3}$ .

Prove the given statement.

Assume an elementx'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 extend to both Sand

S^{c}

.
Hence, 'x' should be included in the boundary of

S^{c}

also.

Thus,

x \in \partial S^{c}

Combine both the statements,

$\partial S = \partial (S^{c})$

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Let S be a subset of $R^{2}$ or $R^{3}$ . Prove that $\partial S = \partial (S^{c})$

Short Answer

Step by step solution

Step 1. Given information.

Prove the given statement.

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Let S be a subset of $R^{2}$ or $R^{3}$ . Prove that $\partial S = \partial (S^{c})$