Question: In Exercise 4, determine whether each set is open or closed or neither open nor closed.

4. a. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} = {\bf{1}}} \right\}\)

b. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} > {\bf{1}}} \right\}\)

c. \(\left\{ {\left( {x,y} \right):{x^{\bf{2}}} + {y^{\bf{2}}} \le {\bf{1}}\,\,\,and\,\,y > {\bf{0}}} \right\}\)

d. \(\left\{ {\left( {x,y} \right):y \ge {x^{\bf{2}}}} \right\}\)

e. \(\left\{ {\left( {x,y} \right):y < {x^{\bf{2}}}} \right\}\)

(a)

Note that the given set is the set of all points on a circle centred at the origin with radius 1.

Hence it contains all its boundary points.

This implies \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} = 1} \right\}\) is closed.

(b)

It is known that the complement of a closed set is an open set.

From part (a), the complement of \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} = 1} \right\}\) is open. That is, \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} > 1\,\,{\rm{and}}\,\,{x^2} + {y^2} < 1} \right\}\) open.

This implies \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} > 1\,} \right\}\)and \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} < 1\,} \right\}\) are open.

(c)

The given set contains all the points of the upper half of the circle centred at the origin with radius one and above the x-axis. Hence it does not include all its boundary points, and it is not open as well.

This implies \(\left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1\,\,{\rm{and}}\,y > 0} \right\}\) is neither open nor closed.

(d)

The given set contains all its boundary points.

Hence, \(\left\{ {\left( {x,y} \right):y \ge {x^2}} \right\}\) is closed.

(e)

It is known that the complement of a closed set is open.

From part (d), the complement of \(\left\{ {\left( {x,y} \right):y \ge {x^2}} \right\}\) is open. That is, \(\left\{ {\left( {x,y} \right):y < {x^2}} \right\}\) open.

Thus, the set is open.