A topological space X is called C-paracompact if there exist a paracompact space Y and a bijective function f : X −→ Y such that the restriction f |A : A −→ f (A) is a homeomorphism for each compact subspace A ⊆ X. A topological space X is called C2-paracompact if there exist a Hausdorff paracompact space Y and a bijective function f : X −→ Y such that the restriction f |A : A −→ f (A) is a homeomorphism for each compact subspace A ⊆ X. We investigate these two properties and produce some examples to illustrate the relationship between them and C-normality, minimal Hausdorff, and other properties.

A C-paracompact is a topological space X associated with a paracompact space Y and a bijective function f : X −→ Y satisfying that f A: A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X. Furthermore, X is called C2-paracompact if Y is T2 paracompact. In this article, we discuss the above concepts and answer the problem of Arhangel’ski ̆i. Moreover, we prove that the sigma product Σ(0) can not be condensed onto a T2 paracompact space.