We introduce new classes of sets called Λg-closed sets and Λg-open sets in topological spaces. We also investigate several properties of such sets. It turns out that Λg-closed sets and Λg-open sets are weaker forms of closed sets and open sets, respectively and stronger forms of g-closed sets and g-open sets, respectively.
Authors:Joao Paulo C. de Jesus and Samuel G. da Silva
With a slight modification of a previous argument due to Schechter, we show that the Axiom of Choice is equivalent to the following topological statement: “If a product of a non-empty family of sets is closed in a topological (Tychonoff) product, then at least one of the factors is closed”. We also discuss the case on which one adds the hypothesis that the closed product of sets is a non-empty set.