A stronger form of locally closed set and its homeomorphic image

Abstract

Through this paper, via the operators $(\cdot)^{\star}$ and $\Psi$, we presented notion of $\star$-Locally set in an ideal topological space $\zeta_{\mathbb{I}}$ as a new stronger form of locally closed set, and considered relations with various existing weak form of locally closed set. Preservations of direct images as well as inverse images of $(\cdot)^{\star}$, $\Psi$, $\star$-perfect and various weak forms of locally closed set including $\star$-Locally closed set are important investigating part. Besides, we pointed out that consideration of `bijectivity' in Lemma 3.1 of [24] is sufficient, and the Lemma 3.3 of [24] is wrong. We demonstrated two modifications of the last one.