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
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“I would have broke mine eye-strings, cracked them, but
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Have turned mine eye and wept.”
—William Shakespeare (1564–1616)