Connectedness and Compactness

A set is connected if it cannot be divided into two disjoint open sets. The set of all rational numbers between 0 and 1 is not connected because it can be divided into two disjoint open sets: the set of all rational numbers less than 1/2 and the set of all rational numbers greater than 1/2. The set of all integers between 0 and 1 is not connected because it is a finite set, and finite sets are not connected. The set of all prime numbers is not connected because it is an infinite set, and infinite sets are not necessarily connected.

A set is compact if it is closed and bounded. The set of all real numbers between 0 and 1 is closed because it contains all of its limit points. It is also bounded because it is contained in the interval [0, 1]. The set of all rational numbers between 0 and 1 is not compact because it is not closed. The set of all integers between 0 and 1 is not compact because it is not bounded. The set of all prime numbers is not compact because it is not bounded.

Every compact set is connected because a compact set cannot be divided into two disjoint open sets. Every connected set is not necessarily compact because a connected set can be unbounded. Every open set is not necessarily connected because an open set can be disconnected. Every closed set is not necessarily compact because a closed set can be unbounded.

The set of all real numbers between 0 and 1 is both connected and compact. It is connected because it cannot be divided into two disjoint open sets. It is compact because it is closed and bounded.

The set of all prime numbers is neither connected nor compact. It is not connected because it can be divided into two disjoint open sets: the set of all prime numbers less than 2 and the set of all prime numbers greater than 2. It is not compact because it is not bounded.