Basic Topology 1 | Introduction and Open Sets in Metric Spaces [dark version]

Topology’s starting point is a shift from measuring distances to describing “closeness” through neighborhoods. The core move begins with metric spaces—sets X equipped with a distance function d—then abstracts away the metric and keeps only the structural behavior of open sets. That matters because many ideas used in analysis (continuity, compactness, convergence) can be reformulated in a way that works even when no explicit distance is available.

A metric space is defined by a function d: X×X → [0,∞) satisfying three rules: positive definiteness (d(x,y)=0 only when x=y), symmetry (d(x,y)=d(y,x)), and the triangle inequality (d(x,z) ≤ d(x,y)+d(y,z)). These axioms justify the geometric intuition that detours can’t shorten the route between points. With this distance in hand, the basic neighborhoods are open ε-balls: for ε>0, the open ball centered at x is the set of points y with d(x,y)<ε. The requirement ε>0 is essential—otherwise the “neighborhood” would collapse.

Open sets in a metric space are then defined using these balls. A subset A⊆X is open if every point x in A has some ε>0 such that the entire open ε-ball around x lies inside A. Visually, this means points on the boundary cannot be included: near any x in A, there is a whole buffer zone of points still staying within A. The allowable ε can depend on x, shrinking as x approaches the boundary.

From there, the discussion turns to the algebra of open sets. The empty set and the whole space X are open by definition. If A and B are open, then their intersection A∩B is open: for any point x in the intersection, both sets contain ε-balls around x, and the smaller of the two balls fits entirely inside A∩B. Unions behave even more simply: the union of two open sets is open because any ε-ball that works for a point in one of the sets also lies within the union. More generally, the union of an arbitrary collection of open sets (indexed by any set I) remains open.

These closure properties—open sets containing ∅ and X, stable under finite intersections and arbitrary unions—are presented as sufficient to characterize what “open” should mean. Crucially, this characterization no longer relies on the metric itself. That sets up the next step: defining topology in a fully abstract setting where distances may not exist, but the open-set behavior still does. The series therefore uses metric spaces as a foundation, then prepares to generalize by turning these open-set axioms into the definition of a topology in the following installment.