The discussion centers on demonstrating that an infinite union of closed sets in the complex plane, \mathbb{C}, is not necessarily closed. A specific example is provided using closed intervals in \mathbb{R}, specifically A_{n}=[1/n,1-1/n], which illustrates that the union of these sets results in an open set. Participants suggest exploring closed balls or closed rectangles in \mathbb{C} to further investigate this property.
PREREQUISITESMathematics students, particularly those studying topology or complex analysis, and educators looking for examples to illustrate properties of closed sets in \mathbb{C>.
autre said:Homework Statement
Show by example that an infinite union of closed sets in \mathbb{C} need not be closed.
The Attempt at a Solution
In \mathbb{R} I know that an infinite union of the closed sets A_{n}=[1/n,1-1/n] is open. Not sure if it works in \mathbb{C} as well.