THE REGULAR OPEN-OPEN TOPOLOGY FOR FUNCTION SPACES

The regular open-open topology, Troo, is introduced, its properties for spaces of continuous functions are discussed, and Troo is compared to Too, the open-open topology. It is then shown that T,.oo on H(X), the collection of all self-homeomorphisms on a topological space, (X, T), is equivalent to the topology induced on H(X) by a specific quasi-uniformity on X, when X is a


INTRODUCTION.
A set-set topology is one which is defined as follows: Let (X,T) and (Y, T') be topological spaces. Let U and V be collections of subsets of X and Y, respectively. Let F C yx, the collection of all functions from X into Y. Define, for U E U and V E V, ( The fact that Too, on H(X), is actually equivalent to the regular-Pervin topology of quasiuniform convergence will be discussed in section 4 along with the topic of quasi-uniform convergence.
The advantage of the regular open-open topology is the set-set notation which provides us with simple notation and, hence, our proofs are more concise than those using the cumbersome notation of the quasi-uniformity.
We assume a basic knowledge of quasi-uniform spaces. An introduction to quasi-uniform spaces may be found in Fletcher and Lindgren's [2] or in Murdeshwar and Naimpally's book [3]. Throughout this paper we shall assume (X, T) and (Y, T) are topological spaces.  THEOREM 1. Let (X, T) be a semi-regular space and F C C(X, Y). If (Y, T')is Ti for 0, 1, 2, then (F, Too) is Ti for 0, 1, 2.
A topology, T', on F C yx is called an admissible (Arens [5]) topology for F provided the evaluation map, E: (F,T') (X,T) (Y,T'), defined by E(f,x) f(x), is continuous. THEOREM 2. If F C C(X, Y) and X is semi-regular, then T, is admissible for F.
Arens also has shown that if T' is admissible for F C C(X, Y), then T' is finer than To. From this fact and Theorem 2, it follows, as it does for Too, that To C T, when X is semi-regular. Let (G, o) be a group such that (G, T) is a topological space, then (G, T) is a topological group provided the following two maps are continuous. (1) m G G G defined by m(g,, g]) g o g and G G defined by (g) g-a. If only the first map is continuous, then we ctil (G,T) a quasi-topological group (Murdeshwar and Naimpally [3]). Note that H(X) with the binary operation o, composition of functions, and identity element e, is a group. It is not difficult to show that if (X, T) is a topological space and G is a subgroup of H(X) then (G, Too) is a quasi-topological group. However, (G, Too) is not always a topological group (Porter,   Let X be a semi-regular space and let (7 be a subgroup 'of H(X). Then (G, Troo) is a topological group.
PROOF. Let X be a semi-regular space and let G be a subgroup of H(X). Let  Topologies are rarely interesting if they are the trivial or discrete topology. We have previously shown (Porter,[4] The above proof, along with the few needed definitions involving Too, is an example of the simplification that the definition of Troo offers over the quasi-uniform definition and notation. 4.

QUASI-UNIFORM CONVERGENCE.
Recall that if Q is a quasi-uniformity on X, then the topology, TQ, on X, which has as its neighborhood base at x, Bx {U[z] U e Q}, is called the topology induced b, Q. The ordered triple (X, Q, To) is called a quasi-uniform space. A topological space, (X, T) is quasi-uniformizable provided there exists a quasi-uniformity, Q, such that T o T. In 1962, Pervin [7] proved that every topological space is quasi-uniformizable by giving the following construction.
Let (X,T) be a topological space. For each O E T, define the set So (Ox O)U((X\O)xX).
Let S {So O E T}. Then S is a subbasis for a quasi-uniformity, P, for X, called the Pervin quasi-uniformity and, as is easily shown, Tp T.
If we use the same basic structure as above but change the subbasis to S {So O is a regular open set then the quasi-uniformity induced will be called the regular-Pervin quasi-uniformity, RP.
If (X, Q) is a quasi-uniform space then Q induces a topology on H(X) called the topology of quasi-uniform convergence w.r.t. Q, as follows: For each set U Q, let us define W(U) {(f,g) H(X) xH(X) (f(z),g(x)) or_ Uforallz e X}. Then, B(Q) {W(U) V e Q} is a basis for Q', the quasi-uniformity of quasi-uniform convergence w.r.t. Q (Naimpally [8]). Let T O. denote the topology on H(X) induced by Q*. T 0. is called the topology of quasi-uniform convergence w.r.t. Qo. If P is the Pervin quasi-uniformity on X, Tp. is the Pervin topology of quasi-uniform conversence and if RP is the regular-Pervin quasi-uniformity on X, then Tp,p is called the regular-Pervin topology of quasi-uniform convergence, Tap  Let (X, T) be a topological space and let G be a subgroup of H(X). Then, Troo=TRp. on G.
ACKNOWLEDGEMENT. The author would like to thank the Committee for Faculty and Curriculum Development at Saint Mary's College for their financial support.