A Hausdorff uniform space is above, such a uniformity can be defined by a " single " pseudometric, which is necessarily a metric if the space is Hausdorff.
12.
The main intuition behind LMNN is to learn a pseudometric under which all data instances in the training set are surrounded by at least k instances that share the same class label.
13.
(This limit exists because the real numbers are complete . ) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0.
14.
Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have d ( x, y ) = 0 for distinct values x \ ne y.
15.
A topological space is said to be a "'pseudometrizable topological space "'if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.
16.
A topological space is said to be a "'pseudometrizable topological space "'if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.
17.
In a pseudometric proximal relator space X, the neighbourhood of a point x \ in X ( denoted by N _ { x, \ varepsilon } ), for \ varepsilon > 0, is defined by
18.
Less trivially, it can be shown that a uniform structure that admits a countable fundamental system of entourages ( and hence in particular a uniformity defined by a countable family of pseudometrics ) can be defined by a single pseudometric.
