The neighborhood system of a point 
in a topological space 
is the family 
of all neighborhoods of
.
It is upward closed, every finite intersection of its members belongs to 
, and every member contains 
. In addition, if 
, then some 
satisfies 
for every 
. Conversely, families satisfying these neighborhood axioms
determine a unique topology on 
.
Neighborhood System
See also
Neighborhood, Neighborhood System Base, TopologyExplore with Wolfram|Alpha
References
Willard, S. General Topology. Reading, MA: Addison-Wesley, 1970.Cite this as:
Weisstein, Eric W. "Neighborhood System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NeighborhoodSystem.html