Euclidean topology

From Wikipedia, the free encyclopedia

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 Sx,y as follows:[1]

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

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).

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, etc., there exist open sets SA and SB with A ⊆ SA and B ⊆ SB such that SA ∩ SB = ∅.[1]

References

  1. ↑ 1.0 1.1 1.2 Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X 
This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.