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We introduce a new class of functions called almost ${\tilde{g}}_{\alpha }$-closed and use the functions to improve several preservation theorems of normality and regularity and also their generalizations. The main result of the paper is that normality and weak normality are preserved under almost ${\tilde{g}}_{\alpha }$-closed continuous surjections.

The aim of the paper is to introduce and study a new class of spaces called mildly ${(1,2)}^{*}$-normal spaces and a new class of functions called ${(1,2)}^{*}$-$\mathrm{rg}$-continuous, ${(1,2)}^{*}$-$\mathrm{R}$-map, almost ${(1,2)}^{*}$-continuous function and almost ${(1,2)}^{*}$-$\mathrm{rg}$-closed function in bitopological spaces. Subsequently, the relationships between mildly ${(1,2)}^{*}$-normal spaces and the new bitopological functions are investigated. Moreover, we obtain characterizations of mildly ${(1,2)}^{*}$-normal spaces, properties of the new bitopological functions and preservation theorems for...