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 Sx,y as follows:
Along with the set R and the empty set ∅, the sets Sx,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 Sx,y for different pairs of (x,y).
Read more about this topic: Euclidean Topology
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