(Proof) Doubts on set of Complex numbers being open and closed

Question

In a course of "Mathematical Methods of Phyisics" (5th semester pregrade course, which basicly consist in Complex Variable) we briefly discussed some topological notions about how to characterize sets. One guide textbook is "Elements of complex variables - Louis Pennisi".

Later, and using some definitions of, we concluded that the Complex numbers set didn't include infinity, therefore, it must not consider the whole boundary of itself (includes none). Also, in the same argument, we considered that the only way of having a set of {} points in the boundary implies that the empty set is included, but since the only element of the set (boundary) is this last one, we could argue that the complex set is also closed (because it includes every element of the boundary).

This is a Physics course so I'm really doubting about the rigour in this logic and proof itself. What do you think?, how would you show this in a formal and "provable" (valid) way?.

Also, is there any set that includes infinity or this is just a boundary concept that implies it will never be considered? In this same reasoning, could I say that every open set that is "not bounded" to other bigger numbers (only by Infinity) is also open and closed? (e.g., real numbers with the infinity in the boundary).

Is this useful in the subject?

Answer 1

It is part of the very definition of a topological space that every topological space is a closed subset of itself and an open subset of itself. With that said:

• The complex plane $\mathbb C$ is a topological space in its own right, and as such is both closed and open in itself.
• The extended complex plane $\mathbb C \cup \{\infty\}$, also known as the Riemann sphere, is also a topological space in its own right, and as such is both closed and open in itself.
• One may regard $\mathbb C$ as a subset of $\mathbb C \cup \{\infty\}$, and in this regard $\mathbb C$ is an open subset, but not a closed subset, of $\mathbb C \cup \{\infty\}$.

There is a general pattern here: given a topological space $X$ and a subset $A \subset X$, one may ask whether $A$ is a closed subset of $X$, and one may ask whether $A$ is an open subset of $X$. What's important here is that these are regarded as properties of $A$ as a subset of $X$.

Answer 2

The only open and closed subsets of a connected topological space are the space itself and the null set, by definition.

The complex plane ($\cong \Bbb R^2$) is connected; as is its one-point compactification ($\cong$ the Riemann sphere).