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:
- (non-negative),
- iff (identity of indiscernibles),
- (symmetry) and
- (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.
Read more about Metric Space: Examples of Metric Spaces, Open and Closed Sets, Topology and Convergence, Types of Maps Between Metric Spaces, Notions of Metric Space Equivalence, Topological Properties, Distance Between Points and Sets; Hausdorff Distance and Gromov Metric, Product Metric Spaces, Quotient Metric Spaces, Generalizations of Metric Spaces
Famous quotes containing the word space:
“The merit of those who fill a space in the world’s history, who are borne forward, as it were, by the weight of thousands whom they lead, shed a perfume less sweet than do the sacrifices of private virtue.”
—Ralph Waldo Emerson (1803–1882)