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 *O _{x}* of it there exists

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*}.

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 *C _{α}*(

*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 *C _{α}*(

*Example 2.* If *α* is the family of all compact subspaces of *X*, then the *α*-topology is called the *compact-open* topology and *C _{α}*(

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 *C _{α}*(

*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 *C _{α}*(

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 *C _{α}*(