538 ARCH. MATH.

H o w m a n y m o n o i d a l c l o s e d s t r u c t u r e s a r e t h e r e in T O P ?

By

GEoac GaEVE*)

The question raised in the heading of this paper has been studied before by several authors. R. Brown [2] for example stated, that there are at ]east ten natural product topologies which are associative, ~in~ura showed in [3] that pointwise convergence is the only symmetric monoidal closed structure on TOP and finally Booth and Tillotson [1] gave four topologies making the topological spaces a monoidal closed category. In the following we shall use the method of the last mentioned authors to show that these results can be enlarged enormously: There are at least as many distinct monoidal closed structures on TOP as there are cardinals, namely a proper class.

Let us first summarize some notions and theorems stated in [1] and [5]. For a class A of topological spaces, A ~ A, a continuous map /: A -> Y and an open subset U of a space Z we define W (/, U) to be the set of all continuous maps g: Y --~ Z with g/(A) c U. The sets W(/, U) form a subbasis of the "A-open topology" on the set C ( Y, Z) of all continuous maps from Y to Z. The corresponding function space will be denoted by C A ( Y , Z ). Of course CA([~, [Z) is an inner hom-functor in TOP.

Now taking m to be a cardinal different from 0 and denoting by M the class of all compact Hausdorff spaces of cardinality less or equal m we get from Theorem 2.6 in [1] the following

1. Theorem. The M-open topology de]ines a monoidal closed structure on TOP.

Finally we need a statement about the existence of compact Hausdorff spaces of arbitrary cardinality:

2. Lemma. For every cardinal m there is a compact Hausdor/] space o] cardinality m.

Proo f . Take X to be a discrete space of cardinality m. I f m is finite, then X is compact (and of course Hausdorff). Assuming X to be infinite the Alexandroff eom- pactification of X yields a compact Hausdorff space of cardinality m.

Now using the terminology of Theorem 1 we obtain the main result:

*) This is a revised version of a paper presented in September 1979. I am grateful to the referee for some corrections and in particular for a hint to the article of Booth and Tillotson [1] accepted for publication some months ago. In that paper independently results are proved, which formerly were part of this paper.

Vol. 34, 1980 How many monoidal closed structures are there in TOP ? 539

3. Theorem. For in/inite cardinals m and n with m < n the h'-open topology is properly [iner than the M-open topology. Thws there is a proper class of monoidal closed structures in TOP.

P r o o f . Of course CN(X, Y) is finer than CM(X, Y), we have to find spaces X, Y such tha t CN(X, Y) and CM(X, Y) are distinct: Take iV to be a compact Hausdorff space of eardinality n (cp. 2) and let S be the two element Sierpinski space, i.e. S = ({0, 1}, {0, {0}, {0, 1}}). Of course CN(N, S) carries the compact open topology, hence it suffices to show, tha t CM (N, S) does not ! Assume C}! (iV, S) has the compact open topology. Then W(idiv, {0}) is open in CM(N, S) and because of

I W(idlv, {0}) I = i

one can ~ t e W(idN, {0}) = FI W(/~, {0}) ~ t h /~: K, ~ N, K~ e )I. Ta~ng p i ~ l p

K :---- ~.J/l(Ki) and ]: K --> N to be the inclusion we get ( ' ] W(/~, {0}) = W(], {0}). ~;=1 i = 1

l~ote tha t K is compact and I K I ~ m < n. Hence there is an open neighbourhood Y of K with Y = b N . Define ]: N - - > S by ] ( Y ) : = 0 and / ( N - - Y ) : = 1. Then / is continuous and ] e W(], {0}) but / ~ W(idlv, {0}). This is a contradiction!

Remark tha t theorem 3 constitutes a great difference between algebraic and topological categories: In the algebraic case we have at most one monoidal closed structure belonging to the forgetful functor (cp. [4]); in the topological case there is at least one structure of this kind, namely tha t of "pointwise convergence" (ep. [4]) and in TOP there is even a proper class of monoidal closed structures.

Relerenees

[1] P. BOOTH and J. T~LOTSO~T, Monoidal closed, cartesian closed and convenient categories of topological spaces. To appear in Pacific J. Math.

[2] R. BROW'S, Ten topologies for X • Y. Quart. J. Math. Oxford Ser. (2), 14, 303--319 (1963). [3] J. ~I-N~U-JaA, Tensorproducts in the category of topological spaces. Comment. Math. Univ.

Carolin. 20, 431--446 (1979). [4] G. GmsvE, M-gesehlossene Kategorien. Thesis, Hagen 1978. [5] G. GREYS, How many monoidal closed structures are there in TOP ? Seminarberichte aus dem

Fachbereich Mathematik der Feruuniversit~t 5, 1979.

Anschrift des Autors:

G. Greve Fernuniversit~t Postfach 940 5800 Hagen West Germany a.t.

Eingegangen am 17. 9. 1979 *)

*) Eine BIeufassung ging am 20. 3. 1980 ein.