Overlapping interval topology explained
In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.
Definition
of the
real number line, the
open sets of the topology are
generated from the half-open intervals
with
and
with
. The topology therefore consists of intervals of the form
,
, and
with
, together with
itself and the empty set.
Properties
Any two distinct points in
are
topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be
separated from any other point in
, making
with the overlapping interval topology an example of a
T0 space that is not a
T1 space.
The overlapping interval topology is second countable, with a countable basis being given by the intervals
,
and
with
and
r and
s rational.
See also
References
