We study the concepts of α-I-continuity and α-I-openness in ideal topological spaces, and obtain several characterizations and some properties of these functions. Also,
we investigate its relationship with other types of functions.
In , Açıkgöz et al. introduced and investigated the notions of w-I-continuous and w*-I-continuous functions in ideal topological spaces. In this paper, we investigate their relationships with continuous and θ-continuous functions.
It is assumed that there is a group of unrelated individuals taken at random from a large population which is exposed to the same time-continuous threat of dying. Accumulated loss of each player increases as the game goes on until at least one participant volunteers to take some extra risk on its own. The risk is taken by a volunteer in order to stop the threat may or may not depend on the time of volunteering. This situation can be modeled as an n-player War of Attrition, which ends when one of the players volunteers. We called this sort of generalization, ieThe (n-player) volunteer dilemmalr. Indeed, a two-player volunteer dilemma is equivalent to the original War of Attrition. It was further assumed that both the risk for the volunteer and the in- tensity of the risk of waiting are time dependent according to some integrable function, this instead of being con- stants as assumed in the original War of Attrition model of Maynard Smith. Necessary and sufficient conditions for a strategy to be a Nash strategy are given. This strategy is characterized by a time-intensity of volunteering. In the stationary case the Nash strategy is proven to be ESS.
We introduce the notions of δ-I- open sets and semi δ-I-continuous functions in ideal topological spaces and investigate some of their properties. Additionally, we obtain decompositions
of semi-I-continuous functions and α-I-continuous functions by using δ-I-open sets.
In 1986, Tong  proved that a function f : (X,τ)→(Y,ϕ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, AI-sets and AI -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, ϕ) is continuous if and only if it is α-I-continuous and AI-continuous.