Openness and Closedness of Integer Set

Determine if the set of integers is open, closed, or compact.

When assessing whether a set is open, closed, or compact in the context of real analysis, it's crucial to rely on definitions from topology. In real analysis, a set is defined as open if for every point within the set, there exists an open interval around it that is entirely contained within the set. Conversely, a set is considered closed if its complement is open or equivalently if it contains all its limit points. Compactness, in the context of the real numbers, specifically relates to whether a set is both closed and bounded, according to the Heine-Borel theorem. With these definitions in mind, analyzing the set of integers provides an interesting insight: although it is obviously unbounded like the entirety of the real numbers, each individual integer does not lie within any sufficiently small open interval around it that lies entirely within the set, making the set of integers not open.

Since the set of integers consists only of isolated points, it is closed as there are no limit points outside of the set itself that could be included. Lastly, due to its lack of boundedness, the set of integers cannot be compact despite being closed, illustrating the independence of the concept of closedness from compactness.