# Metric Space

A metric space is an ordered pair where is a set and is a metric on, i.e., a function

such that for any, the following holds:

1. (non-negative),
2. iff (identity of indiscernibles),
3. (symmetry) and
4. (triangle inequality) .

The first condition follows from the other three, since:

The function is also called distance function or simply distance. Often, is omitted and one just writes for a metric space if it is clear from the context what metric is used.

### Other articles related to "metric space, space, metric spaces":

Generalizations of Metric Spaces - Metric Spaces As Enriched Categories
... Every metric space can now be viewed as a category enriched over Set For each set The composition morphism will be the unique morphism in given from the triangle ...
Completed - Mathematical Completeness
... described equivalently as either the completeness of R as metric space or as a partially ordered set (see below) ... A metric space is complete if every Cauchy sequence in it converges ... See Complete metric space ...
Limit Set - Definition For Iterated Functions
... Let be a metric space, and let be a continuous function ... is here needed, since we have not assumed that the underlying metric space of interest to be a complete metric space ...
Minkowski–Bouligand Dimension - Alternative Definitions
... advantage of using balls rather than squares is that this definition generalizes to any metric space ... definition is "external" — one needs to assume the fractal is contained in a Euclidean space, and define boxes according to the external structure "imposed" by the containing space ... measure the amount of "disorder" in the metric space or fractal at scale, and also measure how many "bits" one would need to describe an element of the metric space or fractal to accuracy ...
Convex Metric Space
... In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints ... Formally, consider a metric space (X, d) and let x and y be two points in X ... A convex metric space is a metric space (X, d) such that, for any two distinct points x and y in X, there exists a third point z in X lying between x and y ...

### Famous quotes containing the word space:

Let the space under the first storey be dark, let the water
lap the stone posts, and vivid green slime glimmer
upon them; let a boat be kept there.
Denise Levertov (b. 1923)