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Chapter 8: Q27E (page 437)

Question: 27. Give an example of a closed subset\(S\)of\({\mathbb{R}^{\bf{2}}}\)such that\({\rm{conv}}\,S\)is not closed.

Short Answer

Expert verified

The set is \(S = \left\{ {\left( {x,y} \right):{x^2}{y^2} = 1,\,\,y > 0} \right\}\).

Step by step solution

01

Assume subset  \(S\) such that \({\rm{conv }}S\) is not closed

One of the possible sets is \(S = \left\{ {\left( {x,y} \right):{x^2}{y^2} = 1,\,\,y > 0} \right\}\). This set is not closed as the equation \({x^2}{y^2} = 1,\,\,\,y > 0\) is of a hyperbola in an upper half-plane that is shown below:

02

Check whether the assumed set \(S\)is in \({\mathbb{R}^{\bf{2}}}\)

The hyperbola\(xy = 1\)opens upwards; that is, it satisfies all values of \(x\) and for \(y > 0\).

So, the set \(S\) is in \({\mathbb{R}^{\bf{2}}}\).

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Most popular questions from this chapter

Question: In Exercise 3, determine whether each set is open or closed or neither open nor closed.

3. a. \(\left\{ {\left( {x,y} \right):y > {\bf{0}}} \right\}\)

b. \(\left\{ {\left( {x,y} \right):x = {\bf{2}}\,\,\,and\,\,{\bf{1}} \le y \le {\bf{3}}} \right\}\)

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