Chapter 12: Q 64. (page 932)

Prove that if S is a closed subset of $R^{2}$ or $R^{3}$ , then $S^{e}$ is an open set. This is Theorem 12.12

Short Answer

Expert verified

It is proved that if Sis a closed subset of $R^{2}$ or $R^{3}$ , then $S^{e}$ is an open set .

Step by step solution

Step 1. Given information.

We have given closed subset $S$ of $R^{2}$ or $R^{3}$ .

Prove the given statement.

The objective is to prove the theorem which states that, "If S is a closed subset of $R^{2} o r R^{3}$ then $S^{e}$ is an open set."

Assume S is a closed subset of $R^{2} o r R^{3}$ .

The definition of a closed set is given as, “A subset of $R^{2}$ or $R^{3}$ is said to be closed, if its complement is an open set."

Thus, from the above definition, S can be termed to be closed, if its complement $S^{e}$ is open. Combining the two statements, it is proved that if S is a closed subset of $R^{2}$ or $R^{3}$ , then $S^{e}$ is an open set .

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Prove that if S is a closed subset of $R^{2}$ or $R^{3}$ , then $S^{e}$ is an open set. This is Theorem 12.12

Short Answer

Step by step solution

Step 1. Given information.

Prove the given statement.

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Prove that if S is a closed subset of R2or R3, then Se is an open set. This is Theorem 12.12

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

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Logic and Functions

Probability and Statistics

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Prove that if S is a closed subset of $R^{2}$ or $R^{3}$ , then $S^{e}$ is an open set. This is Theorem 12.12