Functional Analysis 3 | Open and Closed Sets [dark version]

Metric spaces generalize the familiar idea of “balls” around a point, and that single geometric object drives the definitions of open sets, boundary points, closed sets, and closure. An open ball centered at x with radius ε is the set of all points y in X whose distance from x is less than ε. Using these balls, a subset A of a metric space X is called open if every point x in A has some ε>0 such that the entire ε-ball around x stays inside A—meaning points in A never sit right up against a “border” where nearby points fall outside the set.

Boundary points formalize what it means to be on that border. A point x in X is a boundary point of A if every ε-ball around x contains at least one point from A and at least one point from the complement of A (X\A). Boundary points can lie inside A or outside A; what matters is that no matter how small the neighborhood gets, it straddles membership and non-membership. The collection of all boundary points is denoted ∂A. With this, openness and closedness become mirror conditions: A is open exactly when none of its boundary points lie in A (equivalently, ∂A ⊆ X\A). A set A is closed when its complement X\A is open, which matches the intuition that all boundary points of A must belong to A.

The last key notion is closure, written as \bar{A}. Starting from any subset A, its closure is obtained by adding all boundary points: \bar{A} = A ∪ ∂A. This construction always produces a closed set, and it is the smallest closed set that still contains A—so closure is the “minimal completion” needed to capture limit/border behavior.

Concrete examples in the real line (with the usual distance metric) make the definitions operational. Let X be the set of real numbers between 1 and 3 (including 3) together with all real numbers larger than 4. Consider A = [1,3]. Points strictly less than 3 can fit inside an ε-ball that stays within A, but the point 3 needs care because the surrounding space in X is missing the interval (3,4). Even so, choosing ε small enough ensures the ε-ball around 3 does not reach any points outside A that exist in X, so A is both open and closed in this particular universe X. The same universe-dependent effect explains why openness depends on knowing the ambient space X: the ε-ball around 3 is “one-sided” because points between 3 and 4 are not in X.

A second example uses C = {x in X : 1 ≤ x ≤ 2} with 2 included (in the transcript’s notation, C = [1,2] inside the chosen X). For points below 2, small ε-balls remain entirely within C, so they are not boundary points. At x=2, every ε-ball intersects both C (to the left) and X\C (to the right), so the boundary is ∂C = {2}. Since C already contains its boundary, its closure equals C itself, making C closed. The takeaway is that open/closed/closure are not absolute labels; they are determined by how a set sits inside its metric space and by which points its neighborhoods inevitably touch.