Long line (topology)

The term long straight line ( or Alexandroff - line) referred to in the topology of a topological space, which clearly corresponds to a straight line extended into the Uncountable. Since it locally as the line behaves globally but significantly different from that, they are often used in the topology as a counterexample. It is above all one of the most popular examples of a non- paracompact topological space. In the definition of a manifold one usually requires the existence of a countable base or equivalent conditions (see the second axiom of countability ), if you drop this condition, then the long straight as - are considered manifold without countable basis, sentences such as the embedding theorem of - even differentiable Whitney does not apply to such a variety of course.

Definition

The closed long ray L is defined as the Cartesian product of the smallest uncountable ordinal with the half-open interval, equipped with the lexicographical ordering induced by the order topology. The open long jet called the complement of the origin in the closed long ray.

Inverting the order relation on the open long ray, united this ordered set with the closed long ray as a new parent quantity that each element of the former is smaller than any element of the latter, and then provides them with the order topology, we obtain the long straight. Clearly we then stapled in both directions, an open long beam at the origin.

Properties

• The long straight is a normal space, it is not paracompact.