Félmetrikák

Félmetrikák, sometimes translated as semimetrics, are a mathematical concept that generalizes the notion of a metric. In topology and analysis, a metric on a set X is a function d: X x X -> [0, infinity) that satisfies three properties: non-negativity (d(x,y) >= 0), symmetry (d(x,y) = d(y,x)), and the triangle inequality (d(x,z) <= d(x,y) + d(y,z)). Additionally, a metric requires that d(x,y) = 0 if and only if x = y.

A félmetrika, or semimetric, relaxes the condition that the distance must be zero only when the points are identical. Specifically, a function d: X x X -> [0, infinity) is a félmetrika if it satisfies non-negativity, symmetry, and the triangle inequality, but it is permitted for d(x,y) = 0 even when x is not equal to y. This means that two distinct points can have a distance of zero between them.

The concept of a félmetrika is useful in situations where distinguishing between certain distinct elements is not crucial or is naturally impossible. For example, in some areas of computer science or information theory, two different states or pieces of information might be considered indistinguishable in terms of a particular measure of difference.

Topologically, a félmetrika space (X, d) still defines a topology. The open balls B(x, r) = {y in X | d(x,y) < r} form a basis for this topology. However, the topology induced by a félmetrika may not be Hausdorff. In a Hausdorff space, any two distinct points can be separated by disjoint open sets. Due to the possibility of d(x,y) = 0 for x != y, the open balls around x and y might overlap significantly, potentially preventing their separation.