Properties of domain representations of spaces through dyadic subbases

Abstract

A dyadic subbase S of a topological space X is a subbase consisting of a countable collection of pairs of open subsets that are exteriors of each other. If a dyadic subbase S is proper, then we can construct a dcpo DS in which X is embedded. We study properties of S with respect to two aspects. (i) Whether the dcpo DS is consistently complete depends on not only S itself but also the enumeration of S. We give a characterization of S that induces the consistent completeness of DS regardless of its enumeration. (ii) If the space X is regular Hausdorff, then X is embedded in the minimal limit set of DS. We construct an example of a Hausdorff but non-regular space with a dyadic subbase S such that the minimal limit set of DS is empty.

Existence of strongly proper dyadic subbases

“…The set D S is an algebraic pointed dcpo that is the ideal completion of K S . We quote some results known from [6,7]: If S is a proper dyadic subbase of a regular Hausdorff space X, then X is embedded in the space of minimal elements of D S \ K S . Moreover, if X is compact in addition, then we have a quotient map ρ S : D S \ K S → X and X is homeomorphic to the space of minimal elements of D S \ K S .…”

“…If K S is a conditional upper semilattice with least element (cusl), then the domain representation is admissible ( [6]). Whether K S is a cusl depends not only on the subbase {S a n : n < ω, a ∈ {0, 1}} itself but also on the enumeration of S. It has been proved that K S is a cusl regardless of the enumeration of S if and only if S is strongly proper ( [7]). We refer the reader to [1] for the notion of domain representations of topological spaces.…”

“…If S is proper and K S is a conditional upper semilattice with least element (cusl), then we obtain an admissible domain representation of X [6]. Whether K S is a cusl depends not only on S itself but also on the enumeration of S. It has been proved that K S is a cusl regardless of the enumeration of S if and only if S is strongly proper [7]. The definitions of proper and strongly proper dyadic subbases will be recalled in the next section.…”

Computable dyadic subbases and $\mathbf{T}^ω$-representations of compact sets

“…In order to obtain a result akin to Corollary 10 for a larger class of spaces, we will utilize the notion of a proper dyadic subbase. It was introduced in [29], and further studied in [20,30,33,31]. The original motivation was to generalize the role of the binary and signed binary representations of real number: A proper dyadic subbase induces both (1) a "tiling" coding generalizing the binary expansion and (2) "covering" coding (which forms an admissible representation) generalizing the signed binary expansion.…”

Domain Representations Induced by Dyadic Subbases

“…It is shown in [21] that there is a Hausdorff space X and an independent subbase S of X such that D S is equal to T ω and therefore LpD S q does not have enough minimal elements. for all n P N, we get LpD H 1 q " LpD H q.…”