Interior, Closure, Boundary, and Convergence

Interior, closure, boundary

Given any set A, the interior int(A) is the set of points that sit comfortably inside A — points a with some ball B(a, r) ⊆ A. It is the largest open set contained in A. The closure cl(A) is the smallest closed set containing A; equivalently, it is A together with all points that A presses up against. The boundary ∂A is what is left when you subtract the interior from the closure: ∂A = cl(A) \ int(A) — the skin of the set, the points that are neither safely inside nor safely outside.

Take A = (0, 1] on the real line. Its interior is (0, 1) — the endpoint 1 has no ball staying inside A. Its closure is [0, 1] — the point 0 is pressed against by points of A as close as we like. Its boundary is the two-point set {0, 1}. A point a is a limit point of A if every ball around it contains a point of A other than a itself; the closure is exactly A together with all its limit points.

Convergence, transplanted

On the real line, x_n → L means: for every epsilon > 0 there is an index N so that |x_n − L| < epsilon for all n ≥ N. To define convergence in a metric space we change exactly one symbol — replace |x_n − L| with d(x_n, L). A sequence (x_n) converges to L in (X, d) if for every epsilon > 0 there is N such that d(x_n, L) < epsilon whenever n ≥ N. Equivalently: x_n is eventually inside every ball around L. The entire epsilon-N machinery survives the move intact.

One inheritance is worth proving: limits are unique. The argument is the classic triangle-inequality squeeze, and it works in any metric space because the only tools it uses are the axioms themselves.

Claim: in any metric space a sequence has at most one limit.

Suppose x_n -> L and x_n -> M.  We show L = M.
Fix any epsilon > 0.

From x_n -> L: there is N1 with d(x_n, L) < epsilon/2 for n >= N1.
From x_n -> M: there is N2 with d(x_n, M) < epsilon/2 for n >= N2.

Pick any n >= max(N1, N2).  By the triangle inequality,

      d(L, M) <= d(L, x_n) + d(x_n, M)
              <  epsilon/2 + epsilon/2
              =  epsilon.

So d(L, M) < epsilon for EVERY epsilon > 0.
The only non-negative number smaller than every epsilon is 0,
so d(L, M) = 0, and by the identity axiom L = M.   QED

Uniqueness of limits — pure triangle inequality, valid in every metric space.

Closed sets, sequentially

Convergence gives a second, often more usable description of closedness. A set F is closed if and only if it is sequentially closed: whenever a sequence of points of F converges, its limit also lies in F. This is why “closed” feels like “contains its own limits.” To show a set is closed, a clean strategy is to grab a convergent sequence inside it and prove the limit cannot escape.