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Balls, Open Sets, and Closed Sets

From the metric alone we build open and closed balls, then the open and closed sets they generate — the vocabulary of nearness that powers everything after.

Balls as neighborhoods

Fix a point a and a radius r > 0. The open ball B(a, r) is the set of all points strictly closer than r to a: B(a, r) = { x in X : d(x, a) < r }. The closed ball allows equality: B[a, r] = { x : d(x, a) ≤ r }. On the real line B(a, r) is the open interval (a − r, a + r); in the plane with the Euclidean metric it is a round disk; with the max metric the very same definition produces a square. The shape depends on the metric, but the role is always the same: a ball is a precise version of nearby.

Open sets

A set U is open if around every one of its points there is some open ball entirely inside U. Formally: for each a in U there exists r > 0 with B(a, r) ⊆ U. The radius is allowed to depend on the point — points near the “edge” of U get small balls. Openness means: no point of U sits flush against the outside; everyone has room to wiggle.

The name is honest: an open ball is open. This is exactly where the triangle inequality earns its keep. Take a point x inside B(a, r); it sits at distance d(x, a) < r from the center, leaving a sliver of slack r − d(x, a). A ball of that radius around x stays inside the big ball.

Claim: every open ball B(a, r) is an open set.

Let x be an arbitrary point of B(a, r), so d(x, a) < r.
Define the slack s = r - d(x, a).  Since d(x, a) < r, we have s > 0.

We show B(x, s) is contained in B(a, r).
Let y be any point of B(x, s), so d(y, x) < s.
Apply the triangle inequality with center a:

      d(y, a) <= d(y, x) + d(x, a)
              <  s        + d(x, a)
              =  (r - d(x, a)) + d(x, a)
              =  r.

Hence d(y, a) < r, so y is in B(a, r).
Thus B(x, s) is a subset of B(a, r).

Since x was arbitrary in B(a, r), every point has such a ball,
so B(a, r) is open.   QED
Open balls are open — the slack r − d(x, a) is exactly the radius that works.

Closed sets and the rules they follow

A set F is closed when its complement X \ F is open. Closed sets are exactly the ones that contain all their own limits — we make that precise in the next guide. Beware a common trap: open and closed are not opposites. In any space the whole set X and the empty set are both open and closed at once; the interval [0, 1) on the line is neither.

  1. The empty set and the whole space X are open.
  2. Any union (even infinitely many) of open sets is open.
  3. A finite intersection of open sets is open — but an infinite intersection may fail (e.g. the balls B(0, 1/n) intersect to the single point {0}, which is not open on the line).
  4. By complementation the closed sets satisfy the mirror rules: arbitrary intersections and finite unions of closed sets are closed.