In this paper, we have introduce a new open sets in topological spaces named ic-open sets. We present the relation of ic-open sets with other classes of open sets namely : semi-open, and i-open sets. Furthermore, we introduce the notion of ic-continuous map on topological spaces and we prove some properties and characterizations. Also, we study some separation axioms of this new class. Finally, depending on ic-open sets we define a new class called icc-open set and we discuss the same results obtained by the first one, as each open (closed) set in a topological spaces of any kind (X, τ) is icc-open (icc-closed) set.