Worry-free straight

The Sorgefrey line is an example from the mathematical branch of topology named after the mathematician Robert Henry Sorgefrey .

definition

The Sorgefrey straight line is the topological space that is generated on the set of all half-open intervals as a basis, that is, the open sets of this space are the sets that can be represented as any union of half-open intervals . ${\displaystyle R}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle [a,b)}$ ${\displaystyle [a,b)}$

Remarks

• If the half-open intervals are replaced by an analog construction can be carried out. A space is obtained which is homeomorphic to the Sorgefrey line and is evidently a homeomorphism . ${\displaystyle [a,b)}$ ${\displaystyle (a,b]}$ ${\displaystyle x\mapsto -x}$
• The product is called Sorgefrey level and is also an important example in the topology. ${\displaystyle R^{2}=R\times R}$

Examples of open sets

All sets of form

${\displaystyle (-\infty ,a)=\bigcup _{n=0}^{\infty }[a-n,a)}$

${\displaystyle [a,\infty )=\bigcup _{n=0}^{\infty }[a,a+n)}$

are open. Therefore, the sets are not only open, but also closed , that is , have a base of open-closed sets. ${\displaystyle [a,b)}$ ${\displaystyle [a,b)=\mathbb {R} \setminus ((-\infty ,a)\cup [b,\infty ))}$ ${\displaystyle R}$

Every interval that is open with regard to the Euclidean topology is also open with regard to the topology of the Sorgefrey line, because ${\displaystyle (a,b)}$

${\displaystyle (a,b)=\bigcup _{n=1}^{\infty }[a+{\frac {1}{n}},b)}$.

properties

The Sorgefrey Straight has the following properties: ${\displaystyle R}$

• ${\displaystyle R}$is a perfectly normal room .
• ${\displaystyle R}$has the Lebesgue coverage dimension 0.
• ${\displaystyle R}$is totally incoherent .
• ${\displaystyle R}$is not discrete , because a one-element set does not contain a base set. The topology of the Sorgefrey line is really finer than the Euclidean topology . ${\displaystyle \mathbb {R} }$
• ${\displaystyle R}$is separable ( lies close, because every base set contains a rational number), satisfies the first countability axiom (the sets form a neighborhood basis of ) but not the second countability axiom. ${\displaystyle \mathbb {Q} }$ ${\displaystyle [a,a+{\frac {1}{n}})}$ ${\displaystyle a\in R}$
• ${\displaystyle R}$is not metrizable , because for metric spaces the second axiom of countability follows from the separability.
• ${\displaystyle R}$is paracompact but neither σ-compact nor locally compact .

literature

• Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (=  BI university paperbacks . Volume 121 ). Bibliographisches Institut, Mannheim u. a. 1978, ISBN 3-411-00121-6 .
• Lynn Arthur Steen , J. Arthur Seebach: Counterexamples in Topology . 2nd Edition. Springer, New York a. a. 1978, ISBN 0-387-90312-7 .