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We study topological constructions in the recursion theoretic framework of the lattice of recursively enumerable open subsets of a topological spaceX. Various constructions produce complemented recursively enumerable open sets with additional recursion theoretic properties, as well as noncomplemented open sets. In contrast to techniques in classical topology, we construct a disjoint recursively enumerable collection of basic open sets which cannot be extended to a recursively enumerable disjoint collection of basic open sets whose union is dense inX.

The area of interest of this paper is recursively enumerable vector spaces; its origins lie in the works of Rabin [16], Dekker [4], [5], Crossley and Nerode [3], and Metakides and Nerode [14]. We concern ourselves here with questions about maximal vector spaces, a notion introduced by Metakides and Nerode in [14]. The domain of discourse is V∞ a fully effective, countably infinite dimensional vector space over a recursive infinite field F.By fully effective we mean that V∞, under a fixed Gödel numbering, has the following properties:(i) The operations of vector addition and scalar multiplication on V∞ are represented by recursive functions.(ii) There is a uniform effective procedure which, given n vectors, determines whether or not they are linearly dependent (the procedure is called a dependence algorithm).We denote the Gödel number of x by ⌈x⌉ By taking {εn ∣ n > 0} to be a fixed recursive basis for V∞, we may effectively represent elements of V∞ in terms of this basis. Each element of V∞ may be identified uniquely by a finitely-nonzero sequence from F Under this identification, εn corresponds to the sequence whose n th entry is 1 and all other entries are 0. A recursively enumerable (r.e.) space is a subspace of V∞ which is an r.e. set of integers, ℒ(V∞) denotes the lattice of all r.e. spaces under the operations of intersection and weak sum. For V, W ∈ ℒ(V∞), let V mod W denote the quotient space. Metakides and Nerode define an r.e. space M to be maximal if V∞ mod M is infinite dimensional and for all V ∈ ℒ(V∞), if V ⊇ M then either V mod M or V∞ mod V is finite dimensional. That is, M has a very simple lattice of r.e. superspaces.

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