Metrikräume
Metrikräume, also known as metric spaces, are fundamental concepts in mathematics, particularly in topology and analysis. A metric space is a set equipped with a metric, which is a function that defines a distance between each pair of points in the set. This distance function must satisfy certain properties: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.
The formal definition of a metric space is a pair (M, d), where M is a set
1. d(x, y) ≥ 0 for all x, y in M (non-negativity).
2. d(x, y) = 0 if and only if x = y (identity of indiscernibles).
3. d(x, y) = d(y, x) for all x, y in M (symmetry).
4. d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z in M (triangle inequality).
Examples of metric spaces include Euclidean spaces with the standard Euclidean metric, and discrete spaces where
Key concepts in metric spaces include open sets, closed sets, continuity, and convergence. These concepts are