1.1. Definition of α-topologies on C(X, Y). A family α of subsets of a space X is called a network in X if for each point x ∊ X and each neighborhood Ox of it there exists P ∊ α such that x ∊ P ⊂ Ox. This notion was introduced in (Arhangel’skii 1959) in the investigation of metrization conditions for compacta. Each base of X is a network in X; the family of all singletons subsets of X is also a network in X.
A network α is said to be compact if all its elements are compact subspaces of X. We say that α is hereditary if for each P ∊ α any set B ⊂ P that is closed in P also belongs to α.
Since we shall be frequently considering the same set with different topologies, it is convenient to agree on the following notation: Y ≤ X if Y and X are identical as point sets and the topology of X contains the topology of Y, that is, the open sets in Y are also open in X. If U ⊂ X and V ⊂ Y then
[U, V] = {f ∊ C(X, Y) : f(U) ⊂ V}.
Instead of [{x}, V] we shall write [x, V].Let X and Y be topological spaces, and α a network in X. The family {[P, V] : P ∊ α and V is open in Y} is a subbase of a topology on C(X, Y), called the α-topology (and denoted by Ƭ(α, Y) or Ƭ(α)). The space Cα(X, Y) is C(X, Y) together with this topology.
Example 1. If α is the family of all singleton subsets of X, then the α-topology is the topology of pointwise convergence, in this case Cα(X, Y) is denoted by Cp(X, Y).
Example 2. If α is the family of all compact subspaces of X, then the α-topology is called the compact-open topology and Cα(X, Y) is denoted by Ck(X, Y).
The compact-open topology was introduced by Fox and considered by Arens (Arens and Dugundji 1951); (McCoy and Ntantu 1988).
If Y = ℝ, the space of real numbers with the usual topology, then instead of Cα(X, Y), Cp(X, Y) and Ck(X, Y) we write Cα(X), Cp(X) and Ck(X) respectively.
Example 3. A set A ⊂ X is said to be bounded in X if every continuous real-valued function on X is bounded on A. If α is the family of all bounded closed subsets of X, then the α-topology is called the bounded-open topology and instead of Cα(X) = Cα(X, ℝ) we write Cb(X). If X is Dieudonné complete, that is it is complete with respect to the maximal uniform structure compatible with the topology of X (See Engelking 1977), then the only closed bounded subsets of X are the compact sets. On the other hand, a pseudo-compact Tychonoff space is closed and bounded in itself, although it is not necessarily compact (Engelking 1977).
The following varieties of α-topologies are also of interest, although they are not universal as the topology of pointwise convergence and the compact-open topology.
Example 4. If α is the family of all metrizable compact subspaces of X, then instead of Cα(X, Y) and Cα(X) we write Cm(X, Y) and Cm(X). If α is the family of all countably compact subspaces, then Cα(X, Y) and Cα(X) are denoted by Cc(X, Y) and Cc(X), respectively. Spaces of maps with this topology are considered, in particular, in (McCoy and Ntantu 1988).