Some Results On B-I-Open Sets And The Weak Open Mappings

This paper is mainly composed of two parts. In the first part, the author introducesthe new notion of b-I-open sets in the ideal topological spaces and uses b-I-open sets to define b-I-continuous mappings and b-I-compact (b-I-paracompact) spaces. Then some characterizations and several properties are discussed for the new mappings and spaces. In the second part, the ralations between metrizable spaces and g-metrizable spaces are established by weak-open mapping. And this paper gives some internal characterizationsof weak-open, k-images of metric spaces. It is also shown that metrizable spaces or g-metrizable spaces or sn-metrizable spaces or (?)-spaces are preserved by weak-open closed mappings.In the first part, we have the following results:Result 1 (Theorem 2.2) Let (X,τ,I) be an ideal topological space.(1) If U_α∈BIO(X,τ) for eachα∈Δ, then∪{U_α:α∈Δ}∈BIO{X,τ).(2) If A∈BIO(X,τ) and U isα- I-open, then A∩U∈BIO{X,τ).Result 2 (Theorem 2.5) For a mapping f : (X,τ,I)→(Y,σ, J), the followings are equivalent:(1) f is b-I-continuous.(2) For any x∈X and b-I-open V∈(Y,σ,J) containing f(x), there exists a U∈BIO(X,τ) containing x such that f(U)(?)V.(3) For any x∈X and b-I-closed set F in Y not containing f(x), there exists a b-I-closed set H in X not containing x such that f^-1(F)(?) H.Result 3 (Theorem 2.11) b-I-compactness is an invariant of b-I-continuous mapping.In the second part, we have the following results:Result 4 (Theorem 3.1) X is g-metrizable space if and only if it is a weak open, mssc-image of a metric space. Result 5 (Theorem 3.3) For a space X, the followings are equivalent:(1)X is the weak open k-image of a metric space.(2)X is the 1-sequence-covering quotient k-image of a metric space.(3)X has a compact-finte k-closed subsets sequence point-star weak neighborhood network.(4)X is a sequence space which has a compact-finte sn-covers k-closed subsets point-star network.Result 6 (Theorem 3.4) Metrizable spaces or g-metrizable spaces or sn-metrizable spaces or (?) -spaces are preserved by weak open closed mappings.