[Paper Review] Every Separable Metrizable Space has a Proper Dyadic Subbase

Haruto Ohta, Hideki Tsuiki+1 more|arXiv (Cornell University)|May 15, 2013

Advanced Topology and Set Theory3 citations

TL;DR

This paper proves that every separable metrizable space admits a proper dyadic subbase—specifically, a family of regular open sets indexed by ω×2, where each pair (S_{n,0}, S_{n,1}) forms a complementary cover—such that the restriction to the perfect kernel X^♯ forms an independent subbase. The construction ensures closure compatibility (properness) and preserves dimensionality, with the degree of the subbase matching the topological dimension of X.

ABSTRACT

The notions of a proper dyadic subbase and an independent subbase was introduced by H. Tsuiki to investigate in {0, 1, bot}-sequence codings of topological spaces. We show that every separable metrizable space has a proper dyadic subbase whose restriction to the perfect set defined by the Cantor-Bendixson theorem forms an independent subbase of the restricted space.

Motivation & Objective

• To extend the existence of independent subbases from dense-in-itself separable metrizable spaces to all separable metrizable spaces.
• To construct a proper dyadic subbase for any separable metrizable space X such that its restriction to the perfect kernel X^♯ forms an independent subbase.
• To generalize prior results on dimension-preserving subbases by showing that if dim X = m, then there exists a proper dyadic subbase with degree m.

Proposed method

• Leveraging the Cantor-Bendixson decomposition to isolate the perfect kernel X^♯ and the countable scattered remainder X\X^♯.
• Using half-clopen sets in X—regular open sets with boundary contained in X^♯—to extend subbase elements from X^♯ to X while preserving topological properties.
• Applying a recursive inductive construction to define dyadic subbase sets S_{n,i}^* in X, ensuring properness via closure compatibility and independence via non-empty intersections.
• Utilizing the hereditary paracompactness and regularity of separable metric spaces to lift open sets from X^♯ to X with controlled closures.
• Extending the subbase to include a countable base of clopen sets for the 0-dimensional remainder X\X^♯ to ensure full subbase coverage.
• Verifying that the resulting subbase is proper (cl(S(σ)) = cl(S(σ)) for all σ ∈ T^*) and that the degree deg(S) matches dim X = m.

Experimental results

Research questions

• RQ1 Does every separable metrizable space admit a proper dyadic subbase whose restriction to the perfect kernel X^♯ forms an independent subbase?
• RQ2 Can the degree of a proper dyadic subbase be made equal to the topological dimension of the space?
• RQ3 Is it possible to extend an independent subbase on X^♯ to a proper dyadic subbase on the full space X while preserving the degree?
• RQ4 How can the topological structure of X\X^♯ (a countable, 0-dimensional set) be used to complete a dyadic subbase for X?
• RQ5 What conditions ensure that the closure of a subbase intersection S(σ) equals the closure of its corresponding set in the ambient space?

Key findings

• Every separable metrizable space X admits a proper dyadic subbase S* consisting of half-clopen sets in X such that S* restricted to X^♯ forms an independent subbase.
• The construction ensures that cl_X(S(σ)) = cl_X(S*(σ)) for all σ ∈ {0,1,⊥}^ω, satisfying the definition of a proper dyadic subbase.
• If X is a separable metrizable space with dim X = m, then there exists a proper dyadic subbase S* with deg(S*) = m.
• The subbase S* is constructed by lifting a subbase from X^♯ to X using half-clopen sets and extending with a countable base of clopen sets for X\X^♯.
• The method preserves the degree of the subbase, ensuring that deg(S*) = deg(S) where S is an independent subbase on X^♯.
• The result generalizes previous findings by showing that the existence of independent subbases in dense-in-itself spaces extends to all separable metrizable spaces via proper dyadic subbases.