Niemytzki room

The Niemytzki space (after Viktor Vladimirovich Nemytskii ) is a concrete example of a topological space investigated in the mathematical sub-area of topology . A finer topology than the Euclidean topology , the so-called Niemytzki topology, is introduced on the upper half-plane . This creates a topological space that serves as a counterexample in many situations.

The Niemytzki space is also called Niemytzki level or Moore level (after Robert Lee Moore ) by some authors .

definition

Environments in the Niemytzki room

On the upper half-plane , the Niemytzky topology is explained as follows by specifying a surrounding base of the points from X: If and , then be for ${\displaystyle X:=\{(x,y)\in {\mathbb {R} }^{2};y\geq 0\}}$ ${\displaystyle (x_{0},y_{0})\in X}$ ${\displaystyle r>0}$ ${\displaystyle y_{0}>0}$

${\displaystyle U_{r}(x_{0},y_{0}):=\{(x,y)\in X;(x-x_{0})^{2}+(y-y_{0})^{2}<r^{2}\}.}$

Is so be ${\displaystyle y_{0}=0}$

${\displaystyle U_{r}(x_{0},0):=\{(x_{0},0)\}\cup \{(x,y)\in X;(x-x_{0})^{2}+(y-r)^{2}<r^{2}\}.}$

In the case so it is open circle with a radius to which are cut with the upper half-plane is a point on the patch open circle with radius together with this point. ${\displaystyle y_{0}>0}$ ${\displaystyle r}$ ${\displaystyle (x_{0},y_{0})\in X}$ ${\displaystyle U_{r}(x_{0},0)}$ ${\displaystyle (x_{0},0)}$ ${\displaystyle r}$

If one defines a set to be open in the Niemytzki topology, if for every one is having . with the Niemytzki topology is called Niemytzki space. ${\displaystyle V\subset X}$ ${\displaystyle (x_{0},y_{0})\in V}$ ${\displaystyle r>0}$ ${\displaystyle U_{r}(x_{0},y_{0})\subset V}$ ${\displaystyle X}$

Comparison with the Euclidean topology

${\displaystyle (a_{n})_{n}}$converges to , has no limit. ${\displaystyle (0,0)}$ ${\displaystyle (b_{n})_{n}}$

For a point with, the surrounding bases with respect to the Euclidean topology and the Niemytzki topology agree. ${\displaystyle (x_{0},y_{0})\in X}$ ${\displaystyle y_{0}>0}$

A Euclidean neighborhood of a point contains a sufficiently small semicircle around this point. In every such semicircle there is a Niemytzki neighborhood , if one chooses small enough. Conversely, however, no Euclidean neighborhood is contained in a Niemytzki neighborhood of . This shows that the Niemytzki topology is really finer than the Euclidean topology. ${\displaystyle (x_{0},0)}$ ${\displaystyle U_{r}(x_{0},0)}$ ${\displaystyle r}$ ${\displaystyle (x_{0},0)}$

The sequence defined by converges to in both topologies . The sequence defined by converges with respect to the Euclidean topology , but not with respect to the Niemytzki topology; in this the sequence has no limit at all. ${\displaystyle a_{n}:=\left(0,{\frac {1}{n}}\right)}$ ${\displaystyle (a_{n})_{n}}$ ${\displaystyle (0,0)}$ ${\displaystyle b_{n}:=\left({\frac {1}{n}},1-{\sqrt {1-{\frac {1}{n^{2}}}}}\right)}$ ${\displaystyle (b_{n})_{n}}$ ${\displaystyle (0,0)}$ ${\displaystyle (b_{n})_{n}}$

Subspaces

The subspace carries for the subspace topology , the discrete topology . is a closed set with respect to the Niemytzki topology. The subspace topology corresponds to the Euclidean topology. ${\displaystyle X_{0}:=\{(x,0);x\in {\mathbb {R} }\}}$ ${\displaystyle U_{r}(x,0)\cap X_{0}=\{(x,0)\}}$ ${\displaystyle X_{0}}$ ${\displaystyle X\setminus X_{0}}$

Topological properties

The Niemytzki space has a number of topological properties that serve as counterexamples in many situations.

Local compactness

One can show that the Niemytzki space is not locally compact . Nevertheless, it is a closed subspace such that and both are locally compact. ${\displaystyle X_{0}}$ ${\displaystyle X_{0}}$ ${\displaystyle X\setminus X_{0}}$

Separation axioms

The Niemytzki room X is completely regular . In order to separate a closed set from a point outside of it, one needs, in addition to the continuous functions related to the Euclidean topology, also continuous functions related to the Niemytzki topology, functions of the kind

${\displaystyle f_{r,x_{0}}(x,y)={\begin{cases}{\frac {1}{2ry}}((x-x_{0})^{2}+y^{2})&{\text{wenn }}(x-x_{0})^{2}+(y-r)^{2}\leq r^{2},\,y>0\\0&{\text{wenn }}(x,y)=(x_{0},0)\\1&{\text{sonst}}\end{cases}}}$,

with and , which are also continuous with respect to the Niemytzki topology. ${\displaystyle r>0}$ ${\displaystyle x_{0}\in \mathbb {R} }$

One can show that and are disjoint, closed sets that cannot be separated by open sets , i.e. H. X is not normal . ${\displaystyle A:=\{(x,0);x\in {\mathbb {Q} }\}}$ ${\displaystyle B:=\{(x,0);x\in {\mathbb {R} }\setminus {\mathbb {Q} }\}}$

Separability

The Niemytzki room is separable , in fact it is close to . While in the case of metric spaces separability is inherited to subspaces, the non-separable subspace shows that this is generally not the case (the Sorgefrey level is another example of this type). ${\displaystyle X}$ ${\displaystyle \{(x,y)\in X;x,y\in {\mathbb {Q} }\}}$ ${\displaystyle X}$ ${\displaystyle X_{0}\subset X}$

Countability axiom

The Niemytzki space satisfies the first countability axiom , because the sets form a countable surrounding basis of . One can show that it does not satisfy the second axiom of countability. While the second axiom of countability follows from the separability and the first countability axiom in the case of metric spaces, the Niemytzki space shows that this is generally wrong. ${\displaystyle U_{\frac {1}{n}}(x_{0},y_{0}),\,n\in {\mathbb {N} }}$ ${\displaystyle (x_{0},y_{0})}$

literature

• Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (= BI university paperbacks. 121). Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 .
• Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , example 82