In mathematics, and especially general topology, the Euclidean topology is an example of a topology given to the set of real numbers, denoted by R. To give the set R a topology means to say which subsets of R are "open", and to do so in a way that the following axioms are met:
- The union of open sets is an open set.
- The finite intersection of open sets is an open set.
- The set R and the empty set ∅ are open sets.
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).
- The real line, with this topology, is a T5 space. Given two subsets, say A and B, of R with A ∩ B = A ∩ B = ∅, where A denotes the closure of A, etc., there exist open sets SA and SB with A ⊆ SA and B ⊆ SB such that SA ∩ SB = ∅.