Basic Properties of Closure Spaces

This technical report summarized facts from the basic theory of generalized closure spaces and gives detailed proofs for them. Many of the results collected here are well known for various types of spaces. We have made no attempt to find the original proofs. 1. Set-Valued Set-Functions 1.1. Closure, Interior, Neighborhood, and Convergent. In this section, which in part generalizes the results of Day [8], Hammer [18, 12] and Gni lka [13] on extended topologies, we explore the surprising fact that some meaningful topological concepts can already be defined on a set X endowed with an arbitrary set-valued set-function, which we will interpret as a generalized closure operator. More formally, let X be a set, P(X) its power set (i.e., the set of all subsets of X), and let cl : P(X) → P(X) be an arbitrary function. We shall see that it is fruitful to interpret cl as a closure function on X; hence we call cl(A) is the closure of the set A. In order to simplify the notation in the following we write −A instead of X \ A for the complement of A in X. The dual of the closure function is the interior function int : P(X) → P(X) defined by int(A) = −cl(−A) (1) Given the interior function, we obviously recover the closure as cl(A) = −(int(−A)). A set A ∈ P(X) is closed if A = cl(A) and open if A = int(A). In contrast to “classical” topology, open and closed sets will not play a central role in our discussion. Furthermore, we emphasize that the distinction of closure and interior is completely arbitrary in the absence of additional conditions.