Euclidean topology

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:^[1]

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. The set R and the empty set ∅ are open sets.

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:^[1]

$S_{x,y} = \{ r \in \bold{R} : x < r < y \} .$

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 of various sets S_x,y for different pairs of (x,y).

Properties

• 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, there exist open sets S_A and S_B with A ⊆ S_A and B ⊆ S_B such that S_A ∩ S_B = ∅.^[1]

References

1. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X