Blog on Applied Mathematics

Euclidean distance and others: metric spaces and topology [Part 2]

Short sum up...

richiamo_distanza

In a previous post we talked about Euclidean and other distances...

In that post we introduced the definition of Euclidean, taxi driver and infinity distances. Almost anyone of you remembers the minimum distance, at the end of the post we explained that the minimum distance is not acceptable as a measure because it does not fulfill the distance definition requirements.

We can summarize the requirements a distance need to fulfill in order to be acceptable. It should:

  • be positively defined (greater than or equal to 0)
  •  be symmetric
  • respect triangular inequality

Reading again the previous post, a certain curiosity to know more about the distance and the way it influences our life grew in me... In this post I will try to explain the concept of proximity between two geometrical entities.

Excursus on metrical sets

factory

For a moment we come back to the company and the biker of the previous post. We supposed that only one employee uses a bike to go to work, what happens if the number of employees that use bike to go to work is n, big as you want. Everyone of these bikers has a certain distance from the company, and it doesn't matter which definition of distance we use, the important is that it fulfills the distance requirements. We have a lot of couple defined with the notation (B,{n}_{i}) where B is the company position in the space and {n}_{i} on of the n bikers. For each couple we can define a number that represents the distance between the biker and the company.

felixIn maths, the union composed by the company and the bikers is called set. If we add a distance definition to the set we obtain a metrical set. Metrical sets was theorized by Felix Hausdorff at the beginning of 20th century. Hausdorff, when introduced the metrical sets was studying a more general class of sets called topological sets. Metrical sets are a special class of topological ones.

Proximity

In the previous paragraph I introduced in an easy way a very hard concept to understand. What is a topological set? A topological set is a set in which is defined the concept of proximity. Before introduce more complex concepts can be useful to keep in mind two questions, What does it mean proximity? What does it mean that two points are close? From an intuitive point of view (as often happens in maths, intuition drives men before equations...) we agree if I say that two points are close if the distance between them is really small. But... what is the definition of small? A mathematician could answer "as small as I want".

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