On Some Properties of the Space 2*ω of Binary-Digit Sequences
Stathis Livadas
In this article I deal with some computationally interesting properties of the space 2*ω of finite and infinite binary sequences and of its subspace 2ω of maximal points usually taken with spectral or Stone topology. First, I define 2*ω in terms of a metric-induced quasi-uniformity Ud and then based on the more extended notion of strong inclusion relations I prove the completeness of the quasi-uniform space 2*ω as a syntopological space and the coincidence of its quasi-uniform completion with the usual completeness of it as a uniform space; the latter is proved on the grounds of certain results by P. Sünderhauf in [15]. Next, taking the union 2<ω of the spaces 2n of finite binary sequences, for all n ∈ ω, each one taken with the patch of Alexandroff topology I prove that its one-point (Alexandroff) compactification is a retract of the Cantor space 2ω and its base of clopen sets is a projective Boolean algebra with a unique regular open completion. This provides a new approach to the compactness of space 2ω and has to do with some general results on the one-point compactification of injective Boolean spaces in [5].
Keywords: Cantor space, completeness, metric uniformity, one-point compactification, quasi-uniformity, syntopological space, Stone space.