Chapter 12: Q 65. (page 932)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

We have given S be a subset of $R^{2} o r R^{3}$ .

Prove the given statement.

We have given S be a subset of $R^{2} o r R^{3}$ .

Assume an elementx' 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 .

$x \notin d s$

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

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 $\partial S \cap S = \emptyset$ "

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

Short Answer

Step by step solution

Step 1. Given information.

Prove the given statement.

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Let S be a subset of R2or R3. Prove that a set S is open if and only if ∂S∩S=∅

Short Answer

Step by step solution

Step 1. Given information.

Prove the given statement.

One App. One Place for Learning.

Most popular questions from this chapter

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