Comment on “Soft Mappings Space”
One of the celebrated findings obtained in general topology reports that every compact subset of a Hausdorff space is closed. In this investigation, we demonstrate that this finding need not be true via soft topology. In 2014, Ozturk and Bayramov  discussed some properties of a soft Hausdorff space which defined in [2, 3] and its relationships with a soft compactness notion. However, they made an error, as we observe, in [Theorem 34, p.p.5] which investigated a relationship between soft closed set and soft compact Hausdorff space. To illustrate this mistake, we provide a counterexample and then we conclude under what conditions this result can be generalized via soft topology.
We draw the attention of the readers to the fact that there are different kinds of soft Hausdorff spaces introduced in the literature. Some of them depend on the distinct ordinary points (see, for example, [2–4]) and the others depend on the distinct soft points (see, for example, [5–7]).
First of all, we recall the following three definitions which will be needed throughout this manuscript.
Definition 2 (see ). For a soft set over and , we say that if , for some ; and we say that if , for each .
Definition 3 (see ). A soft set over is said to be stable if there exists a subset of such that , for each .
Now, we mention the alleged results [Theorem 34, p.p.5] as originally proposed in .
Theorem 4. Every soft compact subset of a soft Hausdorff space is soft closed.
In what follows, we construct an example to show that the above theorem is not valid in general.
Example 5. Let be a set of parameters and let the set of real numbers be the universe set. Then we show that a collection either such that is finite or is a soft topology on as follows: (i)Since and is finite, then and since , then .(ii)Let . Then we have the following three cases for arbitrary union and for finite intersection :(1), for each , then and . So and .(2), for each , then and . So and .(3), for some , then and [ or ]. So and . Thus is closed under arbitrary soft union and finite soft intersection. Hence is a soft topology. Moreover, it is a soft Hausdorff space. On the other hand, a soft set , which defined as and , is a soft compact subset of . But it is not soft closed.
Remark 6. It should be noted that the given soft topological space above is considered as a version of Fort space via soft topology.
Before we investigate the correct form of [Theorem 34, p.p.5] in , we present the next auxiliary result.
Lemma 7. (i) Let be a stable soft set over . Then if and only if .
(ii) is a stable soft set if and only if is stable.
Theorem 8. Every stable soft compact subset of a soft Hausdorff space is soft closed.
Proof. Suppose that the given condition holds and let . By hypothesis, is stable, we obtain . For each , we obtain . So . Thus there are two disjoint soft open sets and such that and . It follows that forms a soft open cover of . Consequently, . Putting and . Now, and are soft open sets such that . Therefore and this implies that . Since is chosen arbitrary, then is soft open. Hence is soft closed.
Finally, we give the following result.
Theorem 10. Every soft closed subset of a soft Lindelöf space is soft Lindelöf.
Proof. The proof is similar to that of Theorem 33 of .
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
M. E. El-Shafei, M. Abo-Elhamayel, and T. M. Al-shami, “Partial soft separation axioms and soft compact spaces,” Filomat, vol. 32, no. 13, 2018.View at: Google Scholar