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.

in Purchase, New York (U.S.A.)l) g 1. Introduction and notation Let -!if'( V,) denote the modular lattice of recursively enumerable subspaces of a fixed, infinite dimensional vector space (called V,) under the operations of n and +. The structure of 9( V,) has been studied and compared with that of the somewhat similar lattice 8 of recursively enumerable sets of integers under n und U. Initial work in this area is due to RABIN [la], FROHLICH and SHEPHERDSON [5], DEKKER [3], HAMILTON [7], and R. GUHL [B]. The more recent work of METAKIDES and NERODE [12]. CROSSLEY and NERODE [23], REMMEL [15, 161 and KALANTARI [8] form the basis for current research in this area. Our notation is primarily that of ROGERS' book [19] as well as METAXIDES and NERODE [123. We have collected here examples, theorems and constructions concerning the elementary lattice operation + in 2' ( V,) which grew out of our investigations of this lattice. Of course the operation + for vector spaces corresponds in some sense to the operation of union for sets, however, the dissimilarity between the lattice of all vector subspaces of a given space and the lattice of all subsets of a given set is convincing evidence that the effective lattices of r.e. vector spaces and r.e. sets of integers (traditionally denoted by F ) could be quite different. This indeed proved to be the case, as we shall show.In section 1, the definitions and some of the basic results of the theory of recursively enumerable vector spaces are recalled. For further reference, the reader should consult KALANTARI [8] and REMMEL [15], [16]. Sections 2 and 3 contain two types of results.Theorem 3 and 4 delineate how the property of recursivity fails to be preserved under + .Infact, we show that any infinite dimensional recursively enumerable vector space over an infinite field can be written as a direct sum of two recursively enumerable. infinite dimensional subspaces whose domains are recursive sets of Godel numbers. This shows that the most obvious attempt to effectivize the theory of vector spaces by considering spaces whose domains are recursive sets (and whose basic operations are recursive functions) leads to a weak notion. The stronger, more reasonable notion of recursive space, which is due to METAXIDES and NERODE, namely a space which is complemented in 9 ( V _ ) gives much more familiar results in most cases (see [Sl, 1121, and [18]), however we show below that there are two recursive spaces whose direct sum is not a recursive space. Thus the lattice 9 ( V , ) is genuinely a new object whose theory is quite different from that of 8.The second type of result we have is that of splitting theorems (say A is split into B and C if A = B @ C where A , B, and C are vector spaces). The splitting theorems proven about & by SACKS and FRIEDBERG are recalled and the definitions of splitting are generalized to 9 ( V v , ) . In section 2, a special case of SACK'S splitting theorem is proven to illustrate the close recursion-theoretic relationship possible between r.e. spaces l ) This resea...