# Complex variables : open connected sets

1. Aug 27, 2008

### Benzoate

1. The problem statement, all variables and given/known data

Let S be the open set consisting of all points such that |z|<1 or |z-2|<1 . State why S is not connected.

2. Relevant equations

3. The attempt at a solution

According to my complex variables book the definition of a connected set are pairs of points that can be joined by a polygonal line, consisting of a finite number of line segements joined end to end, that lies entirely in S. (Complex variables and applications, Brown).

I guess S is not connected is because both |z| and |z-2| have the same slope and therefore are parallel to each other . Therefore , since both |z| and |z-2| are parallel to each other, line segments are not connected , since parallel lines will not touch each other.

2. Aug 27, 2008

### Dick

Those inequalities don't describe lines. They describe circular discs. |z|<1 is the open unit disc. Want to try rephrasing that explanation?

Last edited: Aug 27, 2008