Defining system of neighbourhoods
From Encyclopedia of Mathematics
of a set in a topological space
Any family of subsets of the space
subject to the following two conditions: a) for every
there is an open set
in
such that
; b) for any open set
in
containing
there is an element
of the family
contained in
.
It is sometimes further supposed that all elements of the family are open sets. A defining system of neighbourhoods of a one-point set
in a topological space
is called a defining system of neighbourhoods of the point
in
.
References
[1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |
Comments
A defining system of neighbourhoods is also called a local base or a neighbourhood base.
How to Cite This Entry:
Defining system of neighbourhoods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defining_system_of_neighbourhoods&oldid=32509
Defining system of neighbourhoods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defining_system_of_neighbourhoods&oldid=32509
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article