Abstract
The different notions of Cauchy sequence and completeness proposed in the literature for quasi-pseudometric spaces do not provide a satisfactory theory of completeness and completion for all quasi-pseudometric spaces. In this paper, we introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any given quasi-pseudometric space. This new completion theory extends the existing completion theory for metric spaces and satisfies the requirements posed by Doitchinov for a nice theory of completeness.
Highlights
A quasi-pseudometric space (X, d) is a set X together with a nonnegative real-valued function d : X × X → R such that, for every x, y, z ∈ X, (i) d(x, x) = 0 and (ii) d(x, y) ≤ d(x, z) + d(z, y)
We introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any given T0 quasi-pseudometric space
We develop a nonsymmetric completion theory which gives a solution to the problem and extends naturally the existing completion theory for metric spaces
Summary
There are various generalizations to the notion of Cauchy sequence, but, up to now, none of these generalizations is able to give a satisfying completion theory for all quasi-pseudometric spaces. This definition is the following: a sequence (xn)n∈N in a quasi-pseudometric space (X, d) is called Cauchy sequence, if for each σ ∈ N there are a yσ ∈ X and a Nσ ∈ N such that d(yσ, xn) < 1/σ when n > Nσ. This κ-cut defines a new point x0 in a complete quasi-pseudometric space (X, d) (see Theorem 31).
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