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We introduce and investigate forms of regularity and normality defined by means of h-open sets. Using the associated topology τh = hO(X), we interpret h-regularity as a relative regularity property in which points and closed sets of the original topology are separated by τh-open sets, whereas strong h-regularity is precisely the ordinary regularity of (X, τh). We establish characterizations of h-regularity, study its preservation under suitable mappings, compare the new notions with their classical counterparts, and examine related separation properties.