General construction of monoidal closed structures in topological, uniform and nearness spaces

Abstract

In the following paper we consider topological structures on function spaces and cartesian products being connected by an exponential law of the form C(X⊙Y,Z) ≅ C(X,C(Y,Z)). Topological categories provided with such a "monoidal closed" structure are suitable base categories for topological algebra, algebraic topology, automata- or duality theory, in particular if ⊙ is symmetric or the usual direct product. We start from a purely categorical point of view proving an extension theorem which later turns out to be very convenient for the construction of monoidal closed structures in concrete categories, namely in topological spaces, uniform spaces, merotopic spaces and nearness spaces. Enlarging a theorem of Booth and Tillotson [2] it is shown that there are arbitrary many (non symmetric) monoidal closed structures in these categories, hence there is a great difference to the symmetric case, where closed structures seem to be unique (cp. Činčura [3], Isbell [8]). A further application of the extension theorem is a criterion for monoidal- resp. cartesian closedness of MacNeille completions. Of course a symmetric monoidal closed structure is uniquely determined by its values on a finally and initially dense subcategory, but also the converse statement is true, i.e. monoidal closed structures can be obtained by extending a suitable structure from a subcategory to its MacNeille completion.