Sandwich type near-rings using some generalized continuities (Q698589)

Let \(X\) be a topological space, \(G\) an additive topological group, and \(\alpha\) a continuous function from \(G\) into \(X\). The collection of all continuous functions from \(X\) to \(G\) is a right nearring when addition of functions is pointwise and the product \(fg\) of two such functions is defined by \(fg=f\circ\alpha\circ g\). This nearring is referred to as a sandwich nearring with sandwich function \(\alpha\) and is denoted by \(N(X,G,\alpha)\). Such nearrings have been investigated by various people including the reviewer. The point of departure for the present author is to consider structures on sets which are not generally topologies but where the functions respect these structures in some manner. We discuss one of these in further detail in order to give an idea of the nature of the paper. Let \((X,\tau)\) be a topological space. A subset \(A\) of \(X\) is defined to be generalized closed or more briefly g-closed if \(\text{cl}(A)\subseteq G\) whenever \(A\subseteq G\) where \(G\) is any open subset of \(X\). A subset \(B\) of \(X\) is defined to be g-open if \(X\setminus B\) is g-closed. Evidently, every closed set is a g-closed set and every open set is a g-open set. A map from a topological space \((X,\tau)\) to a topological space \((Y,\sigma)\) is defined to be g-continuous if the preimage of every closed set in \(Y\) is g-closed in \(X\) and it is said to be a gc-irresolute map if the preimage of every g-closed set in \(Y\) is g-closed in \(X\). In Theorem 3.3, one of the main results here, the author states that if \((X,\tau)\) is a topological space in which any two nontrivial g-open sets are separated and \((G,\oplus,\sigma)\) is a topological group where the g-closed subsets and the closed subsets coincide, then \(G^X_g\) can be made into a sandwich type nearring. This, of course, is true. One need only take the sandwich function to be any constant map from \(G\) to \(X\). What he means to say (and does say in the proof) is that if \(\alpha\) is a gc-irresolute map from \(G\) to \(X\), then \(G^X_g\), the collection of all g-continuous maps from \(X\) to \(G\), is a nearring when addition of functions is pointwise and the product \(fg\) is defined by \(fg=f\circ\alpha\circ g\). Some further observations are in order at this point. The author never defines the term separated but in the proof of Theorem 3.2, it appears that he adopts the usual definition, i.e., the sets \(A\) and \(B\) are separated if \(A\cap\text{cl }B=\emptyset=\text{cl }A\cap B\). If this is the case, this places severe restrictions on the topological space \(X\). Specifically, \(X\) can contain at most two proper open subsets and if it contains two proper open subsets \(G\) and \(H\), then \(G\cap H=\emptyset\) and \(G\cup H=X\).