Basic Topology 1 | Introduction and Open Sets in Metric Spaces

Topology begins with a practical goal: replace “distance” with a more flexible notion of “closeness,” so tools from analysis can work in settings beyond Euclidean space. The starting point is a metric space, where a set X comes with a distance function d that measures how far two points are apart. That distance must be nonnegative, equal to 0 only when the two points coincide (positive definiteness), symmetric (d(x,y)=d(y,x)), and satisfy the triangle inequality (d(x,z) ≤ d(x,y)+d(y,z)). These rules make distance behave predictably, and they also generate the basic topological language of neighborhoods and open sets.

From the metric, the video builds open balls: for a point x and radius ε>0, the open ε-ball consists of all points y with d(x,y)<ε. An open set is then defined in terms of these balls: a subset A of 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. Geometrically, this means points on the boundary cannot be included; near any point of A, there is always a “buffer zone” of points still staying within A. The size of ε may shrink as x approaches the boundary, but it must remain positive for every x in A.

The key shift is that topology does not need the metric explicitly. Instead, it uses the structural properties that open sets satisfy in metric spaces. The empty set and the whole space X are automatically open. If A and B are open, then their intersection is open: for any point x in A∩B, x has an ε-ball contained in A and an ε-ball contained in B, and the smaller of the two balls fits entirely inside the intersection. Similarly, unions preserve openness: if {A_i} is any collection of open sets, then any point in the union lies in some A_i, and the ε-ball that works for that A_i also works for the union. Crucially, this works even for arbitrary (possibly infinite) collections, not just finite ones.

These closure properties—open sets containing ∅ and X, closed under arbitrary unions, and closed under finite intersections—are presented as sufficient to characterize the open-set structure coming from a metric. That observation sets up the general definition of a topology, where one can talk about open sets without ever specifying a distance function. The series positions this open-set framework as the foundation for later topics like closed sets and compactness, and as a bridge from real analysis into more abstract spaces used across mathematics.