Moore plane

The Niemytzki room (after Viktor Vladimirovich Nemytskii ) is an under -studied in the mathematical branch of topology concrete example of a topological space. On the upper half-plane is finer as compared to the Euclidean topology topology, called Niemytzki topology introduced. The result is a topological space, which serves as a counter- example, in many situations.

The Niemytzki room is called by some authors Niemytzki level or Moore - level (by Robert Lee Moore).

Definition

On the upper half-plane, the Niemytzky topology is declared by specifying a basis of neighborhoods of the points of X as follows: If and so is for

If, as is

In case, therefore, is to open circles with radius, which are cut with the upper half-plane is a patch to the point open circle with radius together with this point.

It now defines a set as open in the Niemytzki topology if there is a, for every with. with the Niemytzki topology is called Niemytzki room.

Comparison with the Euclidean topology

For a point with the surrounding bases are consistent with respect to the Euclidean topology and the Niemytzki topology.

A Euclidean neighborhood of a point contains a sufficiently small half-circle around this point. In each such semi-circle a Niemytzki environment is included if you choose small enough. Conversely, however, is not a Euclidean environment in a Niemytzki - around included. This shows that the Niemytzki topology is finer than the real Euclidean topology.

The sequence defined by converges in both topologies. The sequence defined by converges with respect to the Euclidean topology, but not with respect to the Niemytzki topology; this has the consequence is absolutely no limit.

Subspaces

The subspace carries the discrete topology because as a subspace topology. is a closed set with respect to the Niemytzki topology. The sub-space topology to match with the Euclidean topology.

Topological properties

The Niemytzki room has a number of topological properties, which are used in many situations as counter-examples.

Local compactness

It can be shown that the area is not locally Niemytzki compact. Nevertheless, it is a closed subspace such that and are both locally compact.

Separation axioms

The Niemytzki - space X is completely regular. To separate a closed set from a point external you need apart from the Euclidean topology with respect to the continuous functions, also regarding the Niemytzki topology are continuous, yet functions of the type

With and that with respect to the Niemytzki topology are also continuous.

One can show that and are disjoint, closed sets, which may not be separated by open sets, that is, X is not normal.

Separability

The Niemytzki - space is separable, in fact, is dense in. While inherited in the case of metric spaces separability of subspaces, shows the non- separable subspace that this is not true in general ( the Sorgenfrey level is another example of this kind ).

Countable

The Niemytzki - space satisfies the first axiom of countability, as the quantities, form a countable base of neighborhoods. One can show that it does not satisfy the second axiom of countability. While from the separability and the first axiom of countability in the case of metric spaces followed by the second axiom of countability, the Niemytzki room therefore, shows that in general this is false.