A topology T on a nonempty set X is collection of subsets of X having the empty set ∅ and the set X themselves, which is closed under taking arbitrary unio‎n and finite intersection of its elements. Members of a topology on X are called open subsets of X and complements of open subsets of X are called closed subsets of X. A topology on X is called a clopen topology provided that each member of T is both open and closed in X. A function Γ : P(X) → P(X) is called a closure operator (also called a Kuratowski closure operator) if it satisfies the following four properties: • Γ(∅) = ∅; • E ⊆ Γ(E) for every E ⊆ X; • Γ(Γ(E)) = Γ(E) for every E ⊆ X; • Γ(E) ∪ Γ(F) = Γ(E ∪ F) for every E, F ⊆ X. If Γ : P(X) → P(X) is a closure operator, the fixed points of Γ, that is, the collection of all subsets of X such that Γ(E) = E is the collection of closed subsets of some topology on X corresponding the closure operator Γ. Observe that for a mapping f from X to Y , the operator E → f^{-1}(f(E))1 defined on the power set of X, defines a closure operator. Indeed, it is easy to show that the complement of each set E satisfying E = f^{-1}(f(E)) also satisfies this property. In other words, the fixed points of this closure operator are closed subsets corresponding to a clopen topology on X. On the other hand, for every clopen topology T on X one can define a mapping f on X (to some set Y ) such that the topology of X coincides with the topology which is induced by the closure operator corresponding to the mapping f, that is, we have T = {E ⊆ X : E = f^{-1}(f(E))}. In this thesis we consider the clopen topologies on a set X. Indeed, we characterize the clopen topologies on a set X as those that are weak topologies determined by surjective mappings with values in some discrete topological space (that is, spaces whose collection of open sets is the whole power set). In other words, for a given clopen topology on X we find a surjective mapping with values in some discrete topological space such that the given clopen topology on X coincides with the smallest topology on X which makes the mapping f continuous. In addition, we show that a topology admits a clopen base (that is an open base for a topology whose elements are clopen subsets) if and only if it is a weak topology determined by a family of functions with values in discrete spaces.

محمدرضا كوشش خواجويي، مجيد سلامت

مصطفي عين‌اله زاده صمدي

مجيد فخار، علي طاهري‌فر