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Connectedness and Compactness
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Which of the following sets is connected?
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A
The set of all real numbers between 0 and 1.
💡 Explanation:
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.