pszeudometrikák

Pseudometrics are a generalization of metrics in mathematics. A pseudometric on a set X is a function d: X × X → ℝ (the real numbers) that satisfies the following properties for all x, y, z in X:

1. Non-negativity: d(x, y) ≥ 0

2. Identity of indiscernibles: d(x, y) = 0 if and only if x = y. This property is relaxed compared to a metric. In a metric, d(x, y) = 0 implies x = y. For a pseudometric, d(x, y) = 0 does not necessarily mean x = y.

3. Symmetry: d(x, y) = d(y, x)

4. Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z)

The key difference between a pseudometric and a metric is the second property. In a metric, the distance between two distinct points must be strictly positive. In a pseudometric, the distance between two distinct points can be zero. This means that different elements in the set can be considered "indistinguishable" by the pseudometric.

An example of a pseudometric is the zero pseudometric on any set X, defined by d(x, y) = 0 for all x, y ∈ X. Another example is the pseudometric on the set of all functions from [0, 1] to ℝ, where d(f, g) = ∫₀¹ |f(x) - g(x)| dx. This is actually a metric. However, if we consider the pseudometric d(f, g) = |f(0) - g(0)| on the same space of functions, this is a pseudometric but not a metric, as distinct functions can have the same value at x=0.

The concept of pseudometrics is useful in various areas of mathematics, including topology and functional analysis, where it allows for the study of convergence and completeness in spaces that might not satisfy the strict conditions of a metric space. A pseudometric space can be turned into a metric space by considering the quotient space where points are identified if their distance is zero.