Closure of a set, Real Analysis II

Published: 29 August 2024
on channel: Dr. Bevin Maultsby
142
4

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


Watch video Closure of a set, Real Analysis II online without registration, duration hours minute second in high quality. This video was added by user Dr. Bevin Maultsby 29 August 2024, don't forget to share it with your friends and acquaintances, it has been viewed on our site 142 once and liked it 4 people.