According to [P. H. Doyle and J. G. Hocking, Am. Math. Mon. 68, 959–965 (1961; Zbl 0103.15703)], a topological space \(X\) is said to be invertible provided that for every non-empty open subset \(U\) of \(X\), there exists a homeomorphism \(f:X\rightarrow X\) such that the image of the complement of \(U\) in \(X\) under \(f\) lies in \(U\). In particular, \(n\)-spheres (the sets \(\{x\in\mathbb{R}^{n+1}\,|\,\|x\|=1\}\)) are the only invertible \(n\)-manifolds. The new notion at hand, the above paper presented a series of theorems of the following form: if \(X\) is an invertible topological space, and if \(U\) is a non-empty open subset of \(X\), which (as a subspace) has a certain topological property \(P\), then \(X\) itself has the property \(P\). Examples of \(P\) include \(T_0\), \(T_1\), \(T_2\) separation axioms, regularity and normality, separability, axioms of countability, metrizability, and the property of having a compact closure [J. L. Kelley, General topology. 2nd ed. Graduate Texts in Mathematics. 27. New York - Heidelberg - Berlin: Springer-Verlag. (1975; Zbl 0306.54002)]. Additionally, it was shown that, firstly, an invertible space \(X\), which contains an open connected set \(U\), consists of at most two components (which, in case \(X\) is not connected, are the set \(U\) and its complement in \(X\)), and, secondly, every invertible \(T_1\) space, which contains a non-empty open connected set, is connected. Later on, Y. Wong in [Am. Math. Mon. 73, 835–841 (1966; Zbl 0145.19402)] weakened the definition of invertibility, calling a space \(X\) invertible provided that there exists at least one non-empty open subset \(U\) of \(X\) with the above invertibility property. The original definition was then referred to as complete invertibility. Some of the above-mentioned invertibility theorems (e.g., lower separation axioms, countability axioms, as well as compactness of the closure) were shown to be still valid in the relaxed setting.
In [S. C. Mathew, “Invertible fuzzy topological spaces”, STARS: Int. Journal 7, 1–13 (2006)], the concept of (completely) invertible fuzzy topological space was introduced (in the obvious way), which motivated its author (as well as his collaborators) to extend some of the existing crisp invertibility results to the fuzzy setting. Moreover, [S. C. Mathew and A. Jose, Adv. Fuzzy Sets Syst. 5, No. 2, 153–170 (2010; Zbl 1213.54013)] introduced two additional types of invertibility: type 1, which requires an inverting map for a given proper open fuzzy subset of \(X\) to be the identity map on \(X\); and type 2, which requires the identity map on \(X\) to be an inverting map for all invertible open fuzzy subsets (notice that in no way should every open fuzzy subset be invertible). The paper under review continues this line of research, considering relationships between fuzzy compactness (resp. connectedness) and invertibility. Unlike the classical result, the existence of an invertible open fuzzy subset with a compact closure is not enough to guarantee compactness of the whole space [V. Seenivasan and G. Balasubramanian, Ital. J. Pure Appl. Math. 22, 223–230 (2007; Zbl 1172.54310)]. A crisp (or nearly crisp fuzzy) subset with the compact closure, however, does provide compactness of the whole space (Theorems 38, 39 on pp. 540 – 541 of the paper under review). The above connectedness results though can be restored in the fuzzy setting without any additional requirement (Theorems 59, 60 on p. 544).
The paper is relatively well written (the number of typos is not that big) and contains (almost) all preliminaries, which enables the reader to easily follow the developments of the manuscript, whereas the obtained results themselves would certainly benefit both fuzzy and crisp topologists.