Closure Spaces of Finite Type

As well known in a closure space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(M, \mathfrak{D})}$$\end{document} satisfying the exchange axiom and the finiteness condition we can complete each independent subset of a generating set of M to a basis of M (Theorem A) and any two bases have the same cardinality (Theorem B) (cf. [1,3,4,7]). In this paper we consider closure spaces of finite type which need not satisfy the finiteness condition but a weaker condition (cf. Theorem 3.5). We prove Theorems A and B for a closure space of finite type satisfying a stronger exchange axiom. An example is given satisfying this strong exchange axiom, but not Theorems A and B.