Sorgenfrey plane

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

Definition

If the Sorgenfrey straight, so called the Cartesian product with the product topology, the Sorgenfrey level. Here is the Sorgenfrey straight 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.

The Sorgenfrey - level underlying amount is so and the topology of Sorgenfrey level is thus generated by the set of all half-open rectangles of the form as a base.

Examples of open sets

Since the quantities in the Sorgenfrey - lines are open and closed, this also applies. Therefore, the Sorgenfrey level has a base of open - closed sets.

Each with respect to the Euclidean topology open rectangle is open with respect to the topology of the Sorgenfrey level, also because

.

Properties

The Sorgenfrey level has the following characteristics:

  • Is completely regular as a product of a completely regular space.
  • Is totally disconnected.
  • Has the Lebesgue'sche coverage dimension 0
  • Is not discrete, since a one-element set containing no base amount. The topology of the Sorgenfrey level but is strictly finer than the Euclidean topology.
  • Is separable ( is dense, because each base set contains a point with rational coordinates ), 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 abnormal (see below)

Counter-examples

The amount bears as a subspace topology, the discrete topology, since for any point, such as drawing on the left illustrates.

In particular, is not separable with the subspace topology. The Sorgenfrey - level is therefore an example of that separability is generally not inherited by subspaces. Another example of this situation is the Niemytzki space.

As a subset of is completed because the Euclidean topology is already completed with respect. Because of the discreteness of then every subset of completed. Substituting, then, and two disjoint closed sets, which can not be separated by open sets. is not normal. As is Sorgenfrey straight normal, shows the Sorgenfrey level that a product of normal spaces in general is not normal. As is Sorgenfrey straight even paracompact, the Sorgenfrey level is also an example that products of para compact spaces generally are not paracompact again.

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
Lower limit topology Base (topology)
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