Long straight

In topology , the term long straight line (or Alexandroff straight line ) denotes a topological space that clearly corresponds to a straight line that has been lengthened into the uncountable . Since it behaves locally like the straight line, but differs significantly from it globally, it often serves as a counterexample in the topology. Above all, it is one of the most popular examples of non- paracompact topological space. In the definition of a manifold , one usually demands the paracompactness or the existence of a countable basis (the second axiom of countability ); if these conditions are dropped, the long straight line can be viewed as an - even differentiable - manifold without a countable basis, sentences like the embedding theorem of Whitney are of course not valid for such a manifold, because subsets of the Euclidean space are always two-countable (but there is always a smooth embedding in an infinite-dimensional space).

definition

The closed long ray L is defined as the Cartesian product of the smallest uncountable ordinal number with the half-open interval , endowed with the order topology induced by the lexicographical order . The open long ray denotes the complement of the origin in the closed long ray. ${\ displaystyle \ omega _ {1}}$ ${\ displaystyle [0,1)}$ ${\ displaystyle (0,0)}$

If one inverts the order relation on the open long ray, this ordered set is combined with the closed long ray to a new ordered set in such a way that each element of the former is smaller than each element of the latter, and this is then provided with the order topology the long straight . A long, open beam was then clearly attached to the origin in both directions.

properties

The long straight is a normal room , it is not paracompact .
The long straight is compact , but not compact .

literature

Winfried Koch, Dieter Doll : Differentiable structures on manifolds without a countable basis . In: Archives of Mathematics . tape 19 , no. 1 , 1968, p. 95-102 , doi : 10.1007 / BF01898807 .
Hellmuth Kneser , Martin Kneser : Real-analytical structures of the Alexandroff half-line and the Alexandroff line . In: Archives of Mathematics . tape 11 , 1960, pp. 104-106 , doi : 10.1007 / BF01236917 .
Lynn Arthur Steen , J. Arthur Seebach: Counterexamples in Topology . 2nd Edition. Springer, New York NY et al. 1978, ISBN 3-540-90312-7 , pp. 71-72 (New edition: Dover Publications, New York NY 1995, ISBN 0-486-68735-X ).

Individual evidence

^ Rafael Dahmen: Smooth embeddings of the Long Line and other non-paracompact manifolds into locally convex spaces. In: Topology and its Applications No. 202, 2016, pp. 70-79.
↑ Steven G. Krantz : A Guide to Topology (= The Dolciani Mathematical Expositions. 40 = MAA Guides. 4). Mathematical Association of America, Washington DC 2009, ISBN 978-0-88385-346-7 , Chapter 2.10 “Paracompactness”.
↑ Steen, Seebach: Counterexamples in Topology. 1978, p. 172.

[1] Rafael Dahmen: Smooth embeddings of the Long Line and other non-paracompact manifolds into locally convex spaces. In: Topology and its Applications No. 202, 2016, pp. 70-79.

[2] Steven G. Krantz : A Guide to Topology (= The Dolciani Mathematical Expositions. 40 = MAA Guides. 4). Mathematical Association of America, Washington DC 2009, ISBN 978-0-88385-346-7 , Chapter 2.10 “Paracompactness”.

[3] Steen, Seebach: Counterexamples in Topology. 1978, p. 172.