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, τ1) 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, τ1). 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.
Highlights
Borel proved in his 1894 Ph.D. thesis that a countable covering of a closed interval by open intervals has a finite subcover
It turns out that Borel's approach was similar to the approach Heine used to prove in 1872 that a continuous function on a closed interval was uniformly continuous
What if X is not connected? In this case, we look at the connected components of X
Summary
Borel proved in his 1894 Ph.D. thesis that a countable covering of a closed interval by open intervals has a finite subcover. In 1898, Lebesgue (and apparently someone named Cousins in 1895) removed "countable" from the hypothesis of Borel's result. Murugalingam and Lalitha (2010) introduced the concept of Q* sets [2]. Lalitha and Murugalingam (2011) further studied the properties of Q* closed and Q* open sets in affine space [3]. Padma (2015) introduced the concept of Q*O compact spaces and obtained very crucial results [7, 8] and applied results from [6]. Some important results on bitopological spaces are obtained in [1], [4], [5] and [11].
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