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| Within a metric space, a set is '''open''' if every point within the set can be perturbed in any direction and remain within the set. Clearly this is only ever ''not'' the case if a point is precisely on the boundary of a set, so an open set does not include its boundary while a closed set does. Compare to the more fundamental definition of openness in a [[Analysis/TopologicalSpace|topological space]]. | Within a metric space, a set is '''open''' if every point within the set can be perturbed in any direction and remain within the set. Clearly this is only ever ''not'' the case if a point is a [[Analysis/LimitPoint|limit point]] of a set, so an open set does not include any of its limit points while a closed set includes all of them. Compare to the more fundamental definition of openness in a [[Analysis/TopologicalSpace|topological space]]. |
Metric Space
A metric space is a set for which a distance function can be defined.
Not to be confused with a measure space.
Contents
Description
A metric space is a set with a distance (or metric) function as d: M × M -> R.
For any p and q in M, the distance must feature these properties:
d(p,q) ≥ 0, and d(p,q) = 0 only if p = q
Symmetry: d(p,q) = d(q,p)
Within a metric space, a set is open if every point within the set can be perturbed in any direction and remain within the set. Clearly this is only ever not the case if a point is a limit point of a set, so an open set does not include any of its limit points while a closed set includes all of them. Compare to the more fundamental definition of openness in a topological space.