The Study of the Concept of Q*Compact Spaces

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 , and (0, ) A = ∞ then A is not Q*-compact. A subset S of is Q*-compact. Also, if ( , ) X τ is a Q*-compact metrizable space. Then ( , ) X τ is separable. 1 ( , ) Y τ is Q*-compact and metrizable if f is a continuous mapping of a Q*-compact metric space (X, d) onto a Q*-Hausdorff space 1 ( , ) Y τ . 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.


Introduction
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to prove in 1872 that a continuous function on a closed interval was uniformly continuous (actually first proved, but unpublished for 60 years, by Dirichlet in 1852). In 1898, Lebesgue (and apparently someone named Cousins in 1895) removed "countable" from the hypothesis of Borel's result. Thus, the generalized theorem, which is now commonly called the Heine-Borel theorem. 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].  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]. Let ( , ) X τ be a topological space. A subset S in X is said to be Q* closed in ( , ) X τ if S is closed and Int ( ) S φ = . Its compliment ′ is therefore Q* open [9,10]. If every open cover of X has a finite sub cover then X is called a compact space. ( , ) X τ is said to be separable if it has a countable dense subset. Let X be a set and ℑ a family of subsets of X . Then ℑ is said to have Finite Intersection Property if for any finite number 1 2 , ,..., n F F F of members of ℑ , 1 ...

Preliminaries
This section gives an overview of the basic definitions of a compact space, Q * -compact which is the new type of a compact space. Definition Proof: The space [a, b] is homeomorphic to the Q*compact [0, 1] and so, and by , is Q*-compact.
The space ( , ) a b is homeomorphicto (0, ) ∞ . If ( , ) a b were Q*-compact, then (0, ) ∞ would be Q*-compact, but by Theorem: A subset S of ℝ is Q*-compact if and only if S is closed and bounded.
Proof: First suppose that S is Q*-compact. To see that S is bounded is fairly simple: Let Take the complement of the closure of | ( ) : ?
x B x S is covered by a finite subcover of = ℑ .
Since S is closed and bounded, hypothesis tells us that S has both a maximum and a minimum. Let d minS = . Then and this is certainly covered by a finite subcover of ℑ . Therefore, d B ∈ and B is nonempty. If it is shown that B is not bounded above, then it will contain a number p greater than max S. But then, p S S = so we can conclude that S is covered by a finite subcover, and is therefore Q*- , which again contradicts m supB = . Therefore, either way, if B is bounded above, we get a contradiction. We conclude that B is not bounded above, and S must be Q*-compact.
Hence, ( , ) The converse statement is proved similarly. Theorem: Let f be a continuous mapping of a Q*compact metric space (X, d) onto a Q*-Hausdorff space x x ≠ and hence ( ) It is easily verified that 1 d has the other properties required of a metric, and so a metric on Y . Let 2 τ be the topology induced on Y by 1 d . To show that Firstly, by the definition of 1 d , Observe that for a subset C of Y , C is a closed subset of is also Q*-compact since it is also a product of two Q*compact spaces. Conclusively, suppose that the product of any two N Q*compact spaces is Q*-compact. Consider the product N N X X τ τ × × is Q*compact, so the right-hand side is the product of two Q*compact spaces and thus is Q*-compact. Therefore, the lefthand side is also Q*-compact. Proof: Suppose we ensure that χ is not "too large", that is, not "too much larger" than X . X , which implies that X is not connected. This contradicts our assumption, so χ must be connected.

Conclusion
But what if X is not connected? In this case, we look at the connected components of X . Any open set including ∞ must also contain points in each of the components of X (because the complement of the open set is Q*-compact, and if the complement included an entire connected component, then that component would need to be Q*-compact, but it is not). So W contains some points in each of the components. But this would imply that the connected components are not connected, which is our contradiction. So again, χ must be connected.
It is also true that every Q*O compact space is a Q* -Lindelof space. Every Q*O -compact topological space is Q* -countably compact. Since the space is Q*O -compact, every -Q* open covering of X has a finite subcover. Hence, every countable -Q* open covering of X has a finite subcover and therefore it is countably compact.