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It is well known that, although the category of topological spaces is not cartesian closed, it possesses many cartesian closed full subcategories, e.g.: (i) compactly generated Hausdorff spaces; (ii) quotients of locally compact Hausdorff spaces, which form a larger category; (iii) quotients of locally compact spaces without separation axiom, which form an even larger one; (iv) quotients of core compact spaces, which is at least as large as the previous; (v) sequential spaces, which are strictly included in (ii); and (vi) quotients of countably based spaces, which are strictly included in the category (v).We give a simple and uniform proof of cartesian closedness for many categories of topological spaces, including (ii)-(v), and implicitly (i), and we also give a self-contained proof that (vi) is cartesian closed. Our main aim, however, is to compare the categories (i)-(vi), and others like them.When restricted to Hausdorff spaces, (ii)-(iv) collapse to (i), and most non-Hausdorff spaces of interest, such as those which occur in domain theory, are already in (ii). Regarding the cartesian closed structure, finite products coincide in (i)-(vi). Function spaces are characterized as coreflections of both the Isbell and natural topologies. In general, the function spaces differ between the categories, but those of (vi) coincide with those in any of the larger categories (ii)-(v). Finally, the topologies of the spaces in the categories (i)-(iv) are analysed in terms of Lawson duality.MSC (2000): 54D50, 54D55, 54C35, 06B35.Lemma 2·1. For X exponentiable, the evaluation mappingProof. The transpose of E is the identity map on [X ⇒ Y ], hence continuous, and thus E is continuous since X is exponentiable.Lemma 2·2. For X exponentiable, the exponential topology on C(X, Y ) is uniquely determined.Proof. Suppose that [X ⇒ Y ] and [X ⇒ Y ] are C(X, Y ) endowed with topologies that satisfy the properties of an exponential. By Lemma 2·1 the evaluation map E : [X ⇒ Y ] × X → Y is continuous, and then the identity function on C(X, Y ) from [X ⇒ Y ] to [X ⇒ Y ] is continuous since the latter is an exponential. Reversing the roles of [X ⇒ Y ] and [X ⇒ Y ] gives continuity of the identity in the reverse direction.In light of the preceding observation we denote by [X ⇒ Y ] the set C(X, Y ) of continuous maps endowed with the exponential topology.Lemma 2·3. For X exponentiable and g ∈ C(Y, Z), the mapis continuous by Lemma 2·1, and thus its transpose, which one sees directly to be [X ⇒ g], is continuous.It follows from the preceding that every exponentiable X gives rise to a functor [X ⇒ ·] : Top → Top defined by Y → [X ⇒ Y ] and g → [X ⇒ g]. One observes directly from the fact that [X ⇒ Y ] is an exponential that this functor is right adjoint to the functor · × X. Lemma 2·4. The product of two exponentiable spaces is exponentiable.Proof. Let X 1 and X 2 be exponentiable spaces. Then for all spaces A, one has bijections. Hence the exponential topology on C(X 1 , [X 2 ⇒ Y ]) induces an exponential topology on C(X 1 × X 2 , Y ).Given a fa...

We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (single-point intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be considered as "continuous words". Concatenation of continuous words corresponds to refinement of partial information. The usual basic operations cons, head and tail used to explicitly or recursively define functions on words generalize to partial real numbers. We use this fact to give an operational semantics to the above referred extension of PCF. We prove that the operational semantics is sound and complete with respect to the denotational semantics. A program of real number type evaluates to a head-normal form iff its value is different from ⊥; if its value is different from ⊥ then it successively evaluates to head-normal forms giving better and better partial results converging to its value.

Bar recursion arises in constructive mathematics, logic, proof theory and higher-type computability theory. We explain bar recursion in terms of sequential games, and show how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory. In summary, bar recursion calculates optimal plays and optimal strategies, which, for particular games of interest, amount to equilibria. We consider finite games and continuous countably infinite games, and relate the two. The above development is followed by a conceptual explanation of how the finite version of the main form of bar recursion considered here arises from a strong monad of selections functions that can be defined in any cartesian closed category. Finite bar recursion turns out to be a well-known morphism available in any strong monad, specialised to the selection monad.

This article gives an overview of recent work on the theory of selection functions. We explain the intuition behind these higher type objects, and define a general notion of sequential game whose optimal strategies can be computed via a certain product of selection functions. Several instances of this game are considered in a variety of areas such as fixed point theory, topology, game theory, higher type computability and proof theory. These examples are intended to illustrate how the fundamental construction of optimal strategies based on products of selection functions permeates several research areas.