In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.
Given the closed interval
[
−
1
,
1
]
of the real number line, the open sets of the topology are generated from the half-open intervals
[
−
1
,
b
)
and
(
a
,
1
]
with
a
<
0
<
b
. The topology therefore consists of intervals of the form
[
−
1
,
b
)
,
(
a
,
b
)
, and
(
a
,
1
]
with
a
<
0
<
b
, together with
[
−
1
,
1
]
itself and the empty set.
Any two distinct points in
[
−
1
,
1
]
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
[
−
1
,
1
]
, making
[
−
1
,
1
]
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
[
−
1
,
s
)
,
(
r
,
s
)
and
(
r
,
1
]
with
r
<
0
<
s
and r and s rational (and thus countable).
