The Study of the Concept of Q*Compact Spaces

Abstract

The aim of this research is to extend the new type of compact spaces called Q* compact spaces, study its properties and generate new results of the space. It investigate the Q*-compactness of topological spaces with separable, Q*-metrizable, Q*-Hausdorff, homeomorphic, connected and finite intersection properties. The closed interval [0, 1] is Q* compact. So, it is deduced that the closed interval [0, 1] is Q*-compact. For example, if ( X, τ ) = ℝ and A = (0, ∞) then A is not Q*-compact. A subset S of ℝ is Q*-compact. Also, if ( X , τ ) is a Q*-compact metrizable space. Then ( X , τ ) is separable. ( Y , τ 1 ) is Q*-compact and metrizable if f is a continuous mapping of a Q*-compact metric space (X, d) onto a Q*-Hausdorff space ( Y , τ 1 ). An infinite subset of a Q*-compact space must have a limit point. The continuous mapping of a Q*-compact space has a greatest element and a least element. Eleven theorems were considered and their results were presented accordingly.