In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field.
In general, when we define metric space the distance function is taken to be a real-valued function.
The real numbers form an ordered field which is Archimedean and order complete.
These metric spaces have some nice properties like: in a metric space compactness, sequential compactness and countable compactness are equivalent etc.
These properties may not, however, hold so easily if the distance function is taken in an arbitrary ordered field, instead of in
be an arbitrary ordered field, and
a nonempty set; a function
is called a metric on
if the following conditions hold: It is not difficult to verify that the open balls
( x , δ )
form a basis for a suitable topology, the latter called the metric topology on
In view of the fact that
in its order topology is monotonically normal, we would expect
However, under axiom of choice, every general metric is monotonically normal, for, given
is open, there is an open ball
x , δ
Verify the conditions for Monotone Normality.
The matter of wonder is that, even without choice, general metrics are monotonically normal.
is an Archimedean field.
and the trick is done without choice.
Case II:
is a non-Archimedean field.
is open, consider the set
is open, there is an open ball
We would show that with respect to this mu operator, the space is monotonically normal.
Note that
(open set containing
(open set containing
which is impossible since this would imply that either
This completes the proof.