In topological spaces, the study of interior and closure of a set are renowned concepts where the interior is defined as the union of open sets and the closure is defined as the intersection of closed sets. In literature, it is also a significant study while a set is defined as the intersection of open sets, and the union of closed sets. These respective ideas are known as the kernel of a set and its complementary function. Utilizing these ideas, some authors have introduced various kinds of results in topological spaces. Some mathematicians have extended these concepts via Levine's semi-open sets to semi-kernel and its complementary function. The study of these notions is also a remarkable part of the field of topological spaces as the collection of semi-open sets does not form a topology again. In this paper, we have taken the semi-kernel and its complementary function into account to introduce new types of frontier points. After that we have studied and presented several characterizations of these new types of frontiers and established relationships among them. Finally, we have shown that semihomeomorphic images of these new types of frontiers are invariant.

We introduce and study (Lambda, b)-continuous, (Lambda, b)-irresolute and quasi-(Lambda, b)-irresolute mappings. Some characterizations and several properties concerning aforesaid mappings are obtained. The authors also introduce (Lambda, b)-compactness and (\Lambda, b)-connectedness. It is proved that (Lambda, b)-compactness (respectively (\Lambda, b)-connectedness) is preserved under (\Lambda, b)-irresolute mappings. The paper also touches the topics frontier points, Dirichlet's function, filter and algebraic structure of some functions.

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We give a counterexample in this amendment to show that there is an error in consideration of the statement "{\it if $f : X \to Y$ and ${\bf J}$ is an ideal on $Y$, then $f^{-1}({\bf J}) = \{f^{-1}(J) : J \in {\bf J}\}$ is an ideal on $X$}" by Hamlett in his paper "Lindelöf with respect to an ideal" [New Zealand J. Math. 42, 115-120, 2012]. We also modify it here in a new way and henceforth put forward correctly all the results that were based on the said statement derived therein.