on
is a right-order topology induced from
T. [
X] is a quotient space corresponding to
R.
is a quotient topology on [
X].
and
are quotient spaces corresponding to
and
, respectively.
For
, if
, from the definitions of
and
, we have
, i.e., any neighborhood of
a must contain
b and is also the neighborhood of
b. Conversely, any neighborhood of
b must contain
a and is also the neighborhood of
a. Thus, the neighborhood systems of
a and
b are the same. Their common neighborhood system is just the neighborhood system of
a′ on
. In other words, if elements on [
X] having a common neighborhood system are classified as one category, then [
X] is revised as
.
Now we prove that
is compatible. Assuming
and
, from the above discussion,
and
on
have a common neighborhood system.
,
,
a and
b have a common neighborhood system. From the definition of
,
a and
b belong to the same category on
. Thus,
, i.e.,
is compatible.
Finally, we prove that
is the finest one. Assume that
is any partition and
.
We will prove below that
. Let
be a quotient space corresponding to
and its projection be
. For
and
, let
,
.
Assume that
is any neighborhood of
on
. Since
, regarding
as a set on
, it is open, i.e.,
is open on
. Since
,
, i.e.,
is a neighborhood of
x. On the other hand, since
,
x and
y have the same neighborhood system, i.e.,
is also a neighborhood of y. Thus,
. We have
, i.e.,
.