definition of pseudometric
Topics
About: pseudometric space
The table of contents of this article
Starting Context
- The reader knows a definition of set.
Target Context
- The reader will have a definition of pseudometric.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( S\): \(\in \{\text{ the sets }\}\)
\(*dist\): \(S \times S \to \mathbb{R}\)
//
Conditions:
\(\forall s_1, s_2, s_3 \in S\)
(
(
1) \(0 \le dist (s_1, s_2)\)
\(\land\)
(
\(s_1 = s_2\)
\(\implies\)
\(dist (s_1, s_2) = 0\)
)
)
\(\land\)
2) \(dist (s_1, s_2) = dist (s_2, s_1)\)
\(\land\)
3) \(dist (s_1, s_3) \le dist (s_1, s_2) + dist (s_2, s_3)\)
)
//
2: Note
'pseudometric' is different from 'metric' only in that \(dist (s_1, s_2) = 0\) does not imply \(s_1 = s_2\).
