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.
Contents
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
Preliminary definition
Let
-
d ( x , y ) = 0 ⇔ x = y ; -
d ( x , y ) = d ( y , x ) , commutativity; -
d ( x , y ) + d ( y , z ) ≥ d ( x , z ) , triangle inequality.
It is not difficult to verify that the open balls
In view of the fact that
Further properties
However, under axiom of choice, every general metric is monotonically normal, for, given
The matter of wonder is that, even without choice, general metrics are monotonically normal.
proof.
Case I: F is an Archimedean field.
Now, if x in
Case II: F is a non-Archimedean field.
For given
The set A(x, G) is non-empty. For, as G is open, there is an open ball B(x, k) within G. Now, as F is non-Archimdedean,
Now define
If y is not in G(open set containing x) and x is not in H(open set containing y), then we'd show that
From the above, we get that
So we are done!