Families of Darboux functions and topology having ([formula omitted])-property

Abstract

In this note the (J⁎)-property, being a generalization of the (*)-property, (λ*)-property and (d⁎)-property, is defined. There is proved that if J is an ideal of subsets of the real line containing all singletons and a topology τ has the (J⁎)-property then the family of Darboux functions continuous with respect to the topology τ in the domain is strongly porous set in the space of all functions having τ-Darboux property with the supremum metric.