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 (iii).  .

Solution 9 (iii).

See text and/or instructor's solution manual.

    First, observe that

        

The set    is the closed disk of radius 2 centered at the point .

                It is a closed connected region that is bounded.

                It is not open, hence it is not a domain.

        The boundary circle    is included in the set  .

We are done.   

For each    satisfying    there exists an    so that    (choose  ).  These are the points interior to  S.  

If    satisfies    then for any    the epsilon neighborhood    is not entirely contained in  S.  This argument establishes that S is not an open  set.   

Every pair of points in S can be connected with a straight line.  This argument establishes that S is a connected set.

Every point in the boundary of S is also in S.  This argument establishes that S is a closed set.

.  This establishes that S is a bounded set.

This solution is complements of the authors.

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