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.
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