Environment base

In topology , a branch of mathematics, the environment base is a special set system. Special classes of topological spaces such as locally compact spaces and locally convex spaces can be defined via the properties of surrounding bases . In addition, the first axiom of countability draws on the thickness of the surrounding base and thus implies fundamental structural topological properties. An important special case of neighborhood bases are null neighborhood bases .

definition

Given a topological space and in it a . ${\displaystyle (X,\tau )}$ ${\displaystyle x\in X}$

Then a family is called

${\displaystyle {\mathcal {U}}_{x}:=(U_{x,i})_{i\in I}}$

of neighborhoods of an neighborhood basis of , if each neighborhood of supersets is one of the sets for at least one . ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle U_{x,i}}$ ${\displaystyle i\in I}$

Examples

If one considers the , provided with an arbitrary norm , then is ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \|\cdot \|}$

${\displaystyle B_{r}(x):=\{y\in \mathbb {R} ^{n}\,|\,\|x-y\|<r\}}$

the open sphere with radius around the point . An environment basis with regard to the standard topology is then formed by ${\displaystyle r}$ ${\displaystyle x}$

${\displaystyle {\mathcal {U}}_{x}:=\{B_{r}(x)\,|\,r\in (0,\infty )\}}$.

In this case, a countable environment base can also be defined by

${\displaystyle {\mathcal {U}}_{x}:=\{B_{\tfrac {1}{k}}(x)\,|\,k\in \mathbb {N} \}}$.

Analogously, a (countable) environment base can be created in every metric space with regard to the topology generated by the metric over the open spheres ${\displaystyle (X,d)}$

${\displaystyle B_{r}(x):=\{y\in X\,|\,d(x,y)<r\}}$

define.

Special case zero environment base

If there is a topological vector space , an environment base consisting of environments of is also referred to as a zero environment base. For every point and every such zero neighborhood basis one obtains a neighborhood basis of by translation: ${\displaystyle X}$ ${\displaystyle 0\in X}$ ${\displaystyle {\mathcal {U}}_{0}=(U_{0,i})_{i\in I}}$ ${\displaystyle x\in X}$ ${\displaystyle {\mathcal {U}}_{0}}$ ${\displaystyle {\mathcal {U}}_{x}}$ ${\displaystyle x}$

${\displaystyle {\mathcal {U}}_{x}:=x+{\mathcal {U}}_{0}:=(x+U_{0,i})_{i\in I}\;.}$

Related terms

The set of all environments is referred to as the environment filter or environment system of . Hence, the environment filter of is the largest possible environment base of and is a filter according to its name . ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle x}$

properties

If a topological space has at most a countable environment base, then it is said that it satisfies the first countability axiom . From a mathematical point of view, such spaces are "small" and easier to handle.

literature

• Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .
• Horst Schubert : Topology: An Introduction (=  mathematical guidelines ). 4th edition. BG Teubner , Stuttgart 1975, ISBN 3-519-12200-6 .
• Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .