Editing Similarity (geometry) (section)

==Topology==
{{more citations needed section|date=August 2018}}
In [[topology]], a [[metric space]] can be constructed by defining a '''similarity''' instead of a [[distance]]. The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of '''dissimilarity''': the closer the points, the lesser the distance).

The definition of the similarity can vary among authors, depending on which properties are desired. The basic common properties are
# Positive defined:
<math display=block>\forall (a,b), S(a,b)\geq 0</math>
# Majored by the similarity of one element on itself ('''auto-similarity'''):
<math display=block>S (a,b) \leq S (a,a) \quad \text{and} \quad \forall (a,b), S (a,b) = S (a,a) \Leftrightarrow a=b</math>

More properties can be invoked, such as:
* '''Reflectivity''': <math>\forall (a,b)\ S (a,b) = S (b,a),</math> or
* '''Finiteness''': <math>\forall (a,b)\ S(a,b) < \infty.</math> 
The upper value is often set at 1 (creating a possibility for a probabilistic interpretation of the similitude).

Note that, in the topological sense used here, a similarity is a kind of [[measure (mathematics)|measure]]. This usage is '''not''' the same as the ''similarity transformation'' of the {{section link||In Euclidean space}} and {{section link||In general metric spaces}} sections of this article.