Homogeneous Components of a CDH Fuzzy Space

We prove that fuzzy homogeneous components of a CDH fuzzy topological space  (X,T) are clopen and also they are CDH topological subspaces of its 0-cut topological space (X,T0). 

Densely homogeneous fuzzy spaces

We extend the concept of being densely homogeneous to include fuzzy topological spaces. We prove that our extension is a good extension in the sense of Lowen. We prove that a-cut topological space (X,I_a) of a DH fuzzy topological space (X,I) is DH in general only for a=0

Fuzzy n-s-homogeneity and fuzzy weak n-s-homogeneity

Fuzzy n-s-homogeneity and fuzzy weak n-s-homogeneity are introduced in fuzzy bitopological spaces. Several relationships, characterizations and examples related to them are given

Fuzzy Homogeneous Bitopological Spaces

We continue the study of the concepts of minimality and homogeneity in the fuzzy context. Concretely, we introduce two new notions of minimality in fuzzy bitopological spaces which are called minimal fuzzy open set and pairwise minimal fuzzy open set. Several relationships between such notions and a known one are given. Also, we provide results about the transformation of minimal, and pairwise minimal fuzzy open sets of a fuzzy bitopological space, via fuzzy continuous and fuzzy open mappings, and pairwise continuous and pairwise open mappings, respectively. Moreover, we present two new notions of homogeneity in the fuzzy framework. We introduce the notions of homogeneous and pairwise homogeneous fuzzy bitopological spaces. Several relationships between such notions and a known one are given. And, some connections between minimality and homogeneity are given. Finally, we show that cut bitopological spaces of a homogeneous (resp. pairwise homogeneous) fuzzy bitopological space are homogeneous (resp. pairwise homogeneous) but not conversely

On some types of slight homogeneity

As a generalization of the concept SLH space, we introduce the concept of slightly strongly locally homogeneous (SSLH) spaces. Also, we introduce the concepts of slightly dense set as well as slightly separable space, and use them to introduce two new types of slightly countable dense homogeneous spaces. Several results, relationships, examples and counter-examples concerning these concepts are obtained

Soft Homogeneous Components and Soft Products

Firstly, for the topological spaces that contain a minimal open set, we obtain various inclusions between minimal open sets and homogeneity components. For a given soft topological space (X,τ,A), we define soft homogeneous components. We show that soft homogeneous components of (X,τ,A) form a soft partition of the absolute soft set. Also, we show that (X,τ,A) is soft homogeneous if and only if it has only one soft homogeneous component. Moreover, we study the relationships between the soft homogeneous components of (X,τ,A) and the homogeneous component. For the soft topological spaces that contain a minimal soft open set, we obtain various inclusions between minimal soft open sets and soft homogeneity components. In addition, we show that soft homeomorphisms stabilize soft homogeneous components. Additionally, we introduce two soft product theorems concerning soft homogeneity and soft minimality, respectively

Cluster soft sets and cluster soft topologies

The cluster soft point is an attempt to introduce a novel generalization of the soft closure point and the soft limit point. A cluster soft set is defined to be the system of all cluster soft points of a soft set. Then the fundamental properties of cluster soft sets are demonstrated. Moreover, the concept of a cluster soft topology on a universal set is introduced with regard to the cluster soft sets. The cluster soft topology is derived from a soft topology with an associated soft ideal, but it is finer than the original soft topology. On the other hand, if we start constructing the cluster soft topology from another cluster soft topology, we will end up with the first cluster soft topology we started with. The implication of cluster soft topologies is highlighted using some examples. Eventually, we represent the cluster soft closed sets in terms of several forms of soft sets

On Maximal and Minimal Fuzzy Sets in I-Topological Spaces

The notion of maximal fuzzy open sets is introduced. Some basic properties and relationships regarding this notion and other notions of I-topology are given. Moreover, some deep results concerning the known minimal fuzzy open sets concept are given

Three new soft separation axioms in soft topological spaces

Soft $\omega$-almost-regularity, soft $\omega$ -semi-regularity, and soft $\omega$-$T_{2\frac{1}{2}}$ as three novel soft separation axioms are introduced. It is demonstrated that soft $\omega$ -almost-regularity is strictly between "soft regularity" and "soft almost-regularity"; soft $\omega$-$T_{2\frac{1}{2}}$ is strictly between "soft $T_{2\frac{1}{2}}$" and "soft $T_{2}$", and soft $\omega$ -semi-regularity is a weaker form of both "soft semi-regularity" and "soft $\omega$-regularity". Several sufficient conditions for the equivalence between these new three notions and some of their relevant ones are given. Many characterizations of soft $\omega$-almost-regularity are obtained, and a decomposition theorem of soft regularity by means of "soft $\omega$ -semi-regularity" and "soft $\omega$-almost-regularity" is obtained. Furthermore, it is shown that soft $\omega$-almost-regularity is heritable for specific kinds of soft subspaces. It is also proved that the soft product of two soft $\omega$-almost regular soft topological spaces is soft $\omega$-almost regular. In addition, the connections between our three new conceptions and their topological counterpart topological spaces are discussed

Soft Complete Continuity and Soft Strong Continuity in Soft Topological Spaces

In this paper, we introduce soft complete continuity as a strong form of soft continuity and we introduce soft strong continuity as a strong form of soft complete continuity. Several characterizations, compositions, and restriction theorems are obtained. Moreover, several preservation theorems regarding soft compactness, soft Lindelofness, soft connectedness, soft regularity, soft normality, soft almost regularity, soft mild normality, soft almost compactness, soft almost Lindelofness, soft near compactness, soft near Lindelofness, soft paracompactness, soft near paracompactness, soft almost paracompactness, and soft metacompactness are obtained. In addition to these, the study deals with the correlation between our new concepts in soft topology and their corresponding concepts in general topology; as a result, we show that soft complete continuity (resp. soft strong continuity) in soft topology is an extension of complete continuity (resp. strong continuity) in soft topology