Generalised metric

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

R

.

{\displaystyle \scriptstyle \mathbb {R} .}

Preliminary definition

Let

(

F

,

+

,

⋅

,

<

)

{\displaystyle (F,+,\cdot ,<)}

be an arbitrary ordered field, and

M

{\displaystyle M}

a nonempty set; a function

d

:

M

×

M

→

F

+

∪

{

0

}

{\displaystyle d:M\times M\to F^{+}\cup \{0\}}

is called a metric on

M

,

{\displaystyle M,}

if the following conditions hold:

d

(

x

,

y

)

=

0

{\displaystyle d(x,y)=0}

if and only if

x

=

y

{\displaystyle x=y}

;

d

(

x

,

y

)

=

d

(

y

,

x

)

{\displaystyle d(x,y)=d(y,x)}

(symmetry);

d

(

x

,

y

)

+

d

(

y

,

z

)

≥

d

(

x

,

z

)

{\displaystyle d(x,y)+d(y,z)\geq d(x,z)}

(triangle inequality).

It is not difficult to verify that the open balls

B

(

x

,

δ

)

:=

{

y

∈

M

:

d

(

x

,

y

)

<

δ

}

{\displaystyle B(x,\delta )\;:=\{y\in M\;:d(x,y)<\delta \}}

form a basis for a suitable topology, the latter called the metric topology on

M

,

{\displaystyle M,}

with the metric in

F

.

{\displaystyle F.}

Since

F

{\displaystyle F}

in its order topology is monotonically normal, we would expect

M

{\displaystyle M}

to be at least regular.

Further properties

However, under axiom of choice, every general metric is monotonically normal, for, given

x

∈

G

,

{\displaystyle x\in G,}

where

G

{\displaystyle G}

is open, there is an open ball

B

(

x

,

δ

)

{\displaystyle B(x,\delta )}

such that

x

∈

B

(

x

,

δ

)

⊆

G

.

{\displaystyle x\in B(x,\delta )\subseteq G.}

Take

μ

(

x

,

G

)

=

B

(

x

,

δ

/

2

)

.