The axioms formulated by Hausdorff (1919) for his concept of a topological space. These axioms describe the properties satisfied by subsets of elements
in a neighborhood set 
of 
.
1. There corresponds to each point 
at least one neighborhood 
, and each neighborhood 
contains the point 
.
2. If 
and 
are two neighborhoods of the same point 
, there must exist a neighborhood 
that is a subset of both.
3. If the point 
lies in 
,
there must exist a neighborhood 
that is a subset of 
.
4. For two different points 
and 
, there are two corresponding neighborhoods 
and 
with no points in common.
