Closure of a set, Real Analysis II

In this lecture, we define the notion of the closure of a set in a metric space. The closure of a set A is defined as the intersection of all closed sets that contain A. This definition leads to a few key observations: 1. The closure of A is a closed set since it’s defined as the intersection of closed sets. 2. A is a subset of its closure, meaning that A is always contained within its closure. 3. If A is already closed, then A is equal to its own closure. In other words, a set equals its closure if and only if it is closed. 4. The closure of A is the smallest closed set that contains A. (MA 426 Real Analysis II, Lecture 10) To demonstrate these ideas, we worked through a couple of examples. First, we considered the Archimedean set in the real number line, which consists of points of the form 1/n. By analyzing the intersection of closed sets containing this set, we concluded that the closure of the Archimedean set is the set itself together with the point {0}. In a discrete metric space, every set is closed by definition, so the closure of any set in a discrete metric space is the set itself. We then introduced a useful theorem that provides an alternative way to find the closure of a set. This theorem states that the closure of a set A can also be found by taking the union of A with its accumulation points (or limit points). This method can be more practical than using the original definition. We proved this theorem by considering the complement of A and showing that the closure of A’s complement is the complement of the union of A and its accumulation points. Finally, we applied this theorem to quickly compute the closures of several sets: - The closure of the set of rational numbers is the entire real line. - The closure of the open interval (0, 1) is the closed interval [0, 1]. - The closure of the set of points (x, y) in the plane where the product xy is greater than 0 is the set where xy ≥ 0. - The closure of an open disc in the plane plus an additional point on the boundary is the closed disc. #mathematics #maths #math #topology #metricspaces #settheory #ClosedSets #mathlessons #realanalysis #matheducation #advancedcalculus