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Here we have introduced and studied the idea of $ Ig^*$-closed set withrespect to an ideal and investigated some of its properties in Alexandroffspaces. We have also introduced $ Ig^*$-$T_0 $ axiom, $ Ig^*$-$T_1$ axiom, $Ig^*$-$T_\omega $ axiom and explored a relation among them

$ Ig^*$-closed sets and its certain properties in $ \sigma $-spaces with respect to an ideal