Carefree level

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

definition

If the Sorgefrey straight line , then the Cartesian product with the product topology is called the Sorgefrey plane. Here, the lower limit topology one topological space , which on the amount of all the half-open intervals as a base is generated, that is, the open sets of this space are the semi-open intervals as any association reproducible amounts.

The set on which the Sorgefrey plane is based is thus and the topology of the Sorgefrey plane is accordingly generated from the set of all half-open rectangles of the shape as the basis .

Examples of open sets

Since the sets in the Sorgefrey line are open and closed, this also applies to . The Sorgefrey level therefore has a base of open-closed sets.

Every rectangle that is open with respect to the Euclidean topology is also open with respect to the topology of the Sorgefrey plane, because

.

The topology of the Sorgefrey plane is therefore really finer than the Euclidean topology.

properties

The Sorgefrey level has the following properties:

  • is completely regular as the product of a completely regular room .
  • is totally incoherent .
  • has the Lebesgue coverage dimension 0.
  • is not discrete , because a one-element set does not contain a base set. The topology of the Sorgefrey plane is really finer than the Euclidean topology on .
  • is separable ( lies close, because every base set contains a point with rational coordinates), satisfies the first countability axiom (the sets form a neighborhood basis of ), but not the second countability axiom.
  • is not metrizable , because for metric spaces the second axiom of countability follows from the separability.
  • is not a normal room (see below).

Counterexamples

The subspace carries the discrete topology.

The amount bears as a subspace topology , the discrete topology , because for each point is considered as adjacent drawing shows.

In particular, the subspace topology is not separable. The Sorgefrey level is therefore an example that separability is generally not inherited by subspaces. Another example of this is the Niemytzki room .

as a subset of is closed, since it is already closed with respect to the Euclidean topology. Because of the discrete nature of , every subset of is closed in . Substituting so are and two disjoint closed sets that are not open sets separate leave. is therefore not normal. Since the Sorgefrey line is normal, the Sorgefrey plane shows that a product of normal spaces is generally not normal. Since the Sorgefrey line is even paracompact , the Sorgefrey plane is also an example for the fact that products of paracompact rooms are generally not paracompact again.

literature