Complex variables : open connected sets

In summary, the set S is defined as all points that fall within the open circular discs of |z|<1 and |z-2|<1. This set is not connected because the two circular discs have parallel slopes and therefore cannot be joined by a polygonal line. This violates the definition of a connected set, which requires points to be able to be joined by a polygonal line within the set.
  • #1
Benzoate
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Homework Statement



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


Homework Equations





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.
 
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  • #2
Those inequalities don't describe lines. They describe circular discs. |z|<1 is the open unit disc. Want to try rephrasing that explanation?
 
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Related to Complex variables : open connected sets

1. What are complex variables?

Complex variables are mathematical quantities that involve a combination of real and imaginary numbers. They are used to solve problems in various fields such as physics, engineering, and economics.

2. What is an open set in complex variables?

An open set in complex variables is a subset of the complex plane in which any point can be connected to another point within the set by a curve that lies entirely within the set. This means that there are no boundary points in an open set.

3. What does it mean for a set to be connected in complex variables?

A set is considered connected in complex variables if it cannot be divided into two non-empty subsets that are both open and disjoint. In simpler terms, a connected set is one in which all points can be reached from any other point in the set without leaving the set.

4. How are open and connected sets related in complex variables?

In complex variables, an open set must also be connected. This is known as the connectedness property. However, a connected set does not necessarily have to be open.

5. What are some applications of open connected sets in complex variables?

Open connected sets have numerous applications in various fields, such as in the study of functions of a complex variable, conformal mapping, and potential theory. They are also used in the analysis of electric and magnetic fields, fluid dynamics, and signal processing.

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