Lower limit topology

The Sorgenfrey straight is a named after the mathematician Robert Henry Sorgenfrey example from the mathematical branch of topology.

Definition

The Sorgenfrey straight is the one topological space which is generated on the set of all half-open intervals as a basis, that is, the open sets of this space are the representable as arbitrary union of half-open intervals quantities.

Comments

  • Replacing the half-open intervals through, so you can perform an analogous construction. This gives a homeomorphic to Sorgenfrey - line space, is clearly a homeomorphism.
  • The product is called Sorgenfrey level and is also an important example in the topology.

Examples of open sets

All sets of the form

Are open. Therefore, the quantities are not only open, but also to complete because, that is has a base of open - closed sets.

Each with respect to the Euclidean topology open interval is open with respect to the topology of the Sorgenfrey straight, also because

.

Properties

The Sorgenfrey straight has the following properties:

  • Is a perfectly normal room.
  • Has the Lebesgue'sche coverage dimension 0
  • Is totally disconnected.
  • Is not discrete, since a one-element set containing no base amount. The topology of the Sorgenfrey - line, however, is strictly finer than the Euclidean topology.
  • Is separable ( is dense, because each base set contains a rational number ), the first axiom of countability is sufficient ( the sets form a base of neighborhoods ) but not the second axiom of countability.
  • Is not metrizable, because for metric spaces follows from the separability of the second axiom of countability.
  • Is paracompact but not - yet compact locally compact.

Swell

  • Johann Cigler, Hans- Christian Reichel: Topology. A basic lecture. Bibliographical Institute, Mannheim, inter alia, 1978, ISBN 3-411-00121-6 ( BI university paperbacks 121).
  • Lynn Arthur Steen, J. Arthur Seebach: counterexamples in topology. 2nd edition. Springer, Berlin et al 1978, ISBN 0-387-90312-7.
  • Set topology
  • Topological space
Base (topology) Product topology Sorgenfrey plane Paracompact space
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