Generalizations of Metric Spaces
- Every metric space is a uniform space in a natural manner, and every uniform space is naturally a topological space. Uniform and topological spaces can therefore be regarded as generalizations of metric spaces.
- If we consider the first definition of a metric space given above and relax the second requirement, we arrive at the concepts of a pseudometric space or a dislocated metric space. If we remove the third or forth, we arrive at a quasimetric space, or a semimetric space.
- If the distance function takes values in the extended real number line R∪{+∞}, but otherwise satisfies all four conditions, then it is called an extended metric and the corresponding space is called an -metric space. If the distance function takes values in some (suitable) ordered set (and the triangle inequality is adjusted accordingly), then we arrive at the notion of generalized ultrametric.
- Approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances.
- A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and domains.
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Famous quotes containing the word spaces:
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)