Exercise 9.  Consider the following sets.   Sketch each set.   State, with reasons, which of the following terms apply to the above sets: open; connected; domain; region; closed region; bounded.  

9 (vii).  .  

Solution 9 (vii).

See text and/or instructor's solution manual.

    The set    the union of the two open disks

          and  .

                It is an open bounded set.  

                It is not a domain and not a region.

                It is not closed and not bounded.

        The boundary circles   and   are not included in the set  .

We are done.   

For each    there exists an    so that    (if    choose ,  and if    choose ).    This argument establishes that S is an open set.

There exists a pair of points in S that cannot be connected with a curve.  This argument establishes that S is not a connected set.

    .  This establishes that S is a bounded set.

Remark.  The rigorous details for this argument are illustrated in Example 13.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell