Comparison of two unified continuity approaches (Q532103)

The authors call a non-empty collection \(\varepsilon\) of non-empty subsets of a topological space \((X,\tau)\) a cluster system and define a set \(A(\subseteq X)\) to be \(\varepsilon\)-big at a point \(x \in X\) if, for any neighbourhood \(U\) of \(x\), the intersection \(A \cap U\) contains a member of \(\varepsilon\). Then, for a subset \(A\) of \(X\), the set \(\varepsilon_A = \{x \in X : A\;\text{is }\varepsilon\)-big at