Nöbeling room

The Nöbeling space is a construction from the mathematical subfield of topology . It is the universal separable metric space.

It is named after Georg Nöbeling .

construction

The m-dimensional Nöbeling space is the set of all points with at most rational coordinates: ${\displaystyle {\mathcal {R}}_{m}^{2m+1}\subset \mathbb {R} ^{2m+1}}$ ${\displaystyle m}$

${\displaystyle {\mathcal {R}}_{m}^{2m+1}:=\left\{(x_{1},x_{2},\ldots ,x_{2m+1})\in \mathbb {R} ^{2m+1}\colon {\text{Es gibt }}\ 1\leq i_{1}<i_{2}<\ldots <i_{m+1}\leq 2m+1\ {\text{ mit }}x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{m+1}}\not \in \mathbb {Q} \right\}}$.

universality

The m-dimensional Nöbeling space is the universal m-dimensional separable metric space, i.e. H. any m-dimensional separable metric space can be embedded in . ${\displaystyle {\mathcal {R}}_{m}^{2m+1}}$

properties

The m-dimensional Nöbeling space is (m-1) -connected and (m-1) -locally connected. That means

• ${\displaystyle \pi _{i}({\mathcal {R}}_{m}^{2m+1})=0}$for , and ${\displaystyle i\leq m-1}$
• for every neighborhood of a point there is a neighborhood with for . ${\displaystyle U}$ ${\displaystyle x\in {\mathcal {R}}_{m}^{2m+1}}$ ${\displaystyle x\in V\subset U}$ ${\displaystyle \pi _{i}(V)=0}$ ${\displaystyle i\leq m-1}$

Rigidity

Every m-dimensional connected space that is locally too homeomorphic (i.e. for every point there is a neighborhood that is homeomorphic to an open subset of ) is already too homeomorphic. ${\displaystyle {\mathcal {R}}_{m}^{2m+1}}$ ${\displaystyle {\mathcal {R}}_{m}^{2m+1}}$ ${\displaystyle {\mathcal {R}}_{m}^{2m+1}}$

characterization

A topological space X is homeomorphic to the m-dimensional Nöbeling space if it has the following properties:

• X is separable.
• X has a full metric .
• X is m-dimensional.
• X is (m-1) -contiguous.
• X is (m-1) -locally connected.
• X satisfies the locally finite-m-slice property, i.e. H. at any open covering and each episode there is a sequence , so that at any one environment with almost all are and that to each one with there. ${\displaystyle X=\cup U_{i}}$ ${\displaystyle f_{n}\colon D^{m}\to X}$ ${\displaystyle g_{n}\colon D^{m}\to X}$ ${\displaystyle x\in X}$ ${\displaystyle U_{i}}$ ${\displaystyle U_{i}\cap g_{n}(D^{m})=\emptyset }$ ${\displaystyle n}$ ${\displaystyle t\in D^{m}}$ ${\displaystyle U_{i}}$ ${\displaystyle f_{n}(t)\in U_{i},g_{n}(t)\in U_{i}}$

literature

• Andrzej Nagórko: Characterization and topological rigidity of Nobeling manifolds (= Memoirs of the American Mathematical Society. 1048 = 223, 2). American Mathematical Society, Providence RI 2013, ISBN 978-0-8218-5366-5 , arxiv : math.GT/0602574 (dissertation).