Euclidean Topology - Construction

Construction

The set R and the empty set ∅ are required to be open sets, and so we define R and ∅ to be open sets in this topology. Given two real numbers, say x and y, with x < y we define an uncountably infinite family of open sets denoted by S_x,y as follows:

Along with the set R and the empty set ∅, the sets S_x,y with x < y are used as a basis for the Euclidean topology. In other words, the open sets of the Euclidean topology are given by the set R, the empty set ∅ and the unions and finite intersections of various sets S_x,y for different pairs of (x,y).