**Fuzzy topological spaces and fuzzy compactness**

It is the aim of this paper to travel somewhat deeper into the structure of fuzzy topological areas. In doing thus we tend to found we tend to had to change the definition of a fuzzy topology ran down to currently. we tend to shall conjointly introduce 2 functors gw and gi which can permit U.S. to examine additional clearly the association between fuzzy topological areas and topological areas. Finally we tend to shall introduce the construct of fuzzy compactness because the generalization of compactness in topology. it’ll be shown in a very following publication that contrary to the results obtained up to currently, the Tychonoff-product theorem is safeguarded with fuzzy compactness. **[1]**

**Smooth topological spaces**

In 1986, Badard introduced the thought of a sleek space. during this paper we tend to provide some links between sleek topological areas and also the corresponding leads to Chang’s fuzzy topological spaces. The ideas of topological space, continuity and compactness are studied. **[2]**

**On soft topological spaces**

In the gift paper we have a tendency to introduce soft topological areas that square measure outlined over associate degree initial universe with a set set of parameters. The notions of soppy open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of some extent and soft separation axioms square measure introduced and their basic properties square measure investigated. it’s shown that a soft space offers a parametrized family of topological areas. moreover, with the assistance of associate degree example it’s established that the converse doesn’t hold. The soft subspaces of a soft space square measure outlined and inherent ideas also because the characterization of soppy open and soft closed sets in soft subspaces square measure investigated. Finally, soft -spaces and notions of soppy traditional and soft regular areas square measure mentioned well. A adequate condition for a soft space to be a soft -space is additionally conferred. **[3]**

**Identification of Topological Network Modules in Perturbed Protein Interaction Networks**

Biological networks contains useful modules, but detective work and characterizing such modules in networks remains difficult. heavy networks is one strategy for distinctive modules. Here we tend to used a sophisticated mathematical approach named topological knowledge analysis (TDA) to interrogate 2 flustered networks. In one, we tend to noncontinuous the S. cerevisiae INO80 supermolecule interaction network by analytic advancedes once protein complex elements were deleted from the ordering. within the second, we tend to reanalyzed antecedently printed knowledge demonstrating the disruption of the human Sin3 network with a simple protein deacetylase matter. Here we tend to show that noncontinuous networks contained topological network modules (TNMs) with shared properties that mapped onto distinct locations in networks. **[4]**

**Metric on the Countable Soft Topological Space**

The author, essential oil KHARAL, outlined the euclidian distance victimization the bilaterally symmetric distinction of sets in soft area, however, we tend to conclude that every one the sets of “ε-approximate parts among the soft sets cannot calculate their bilaterally symmetric di erence during this manner. for instance our purpose, during this paper, we tend to outline the finite(countable) soft space, and signifies the euclidian distance given by essential oil KHARAL will simply be applied to the calculable soft area. Meanwhile, we tend to offer the overall definition of the metric soft space, check the euclidian distance being a metric during a calculable soft space, and succeed a metric calculable soft topology.** [5]**

**Reference**

**[1]** Lowen, R., 1976. Fuzzy topological spaces and fuzzy compactness. Journal of Mathematical analysis and applications, 56(3), (Web Link)

**[2]** Ramadan, A.A., 1992. Smooth topological spaces. Fuzzy sets and Systems, 48(3), (Web Link)

**[3]** Shabir, M. and Naz, M., 2011. On soft topological spaces. Computers & Mathematics with Applications, 61(7), (Web Link)

**[4]** Identification of Topological Network Modules in Perturbed Protein Interaction Networks

Mihaela E. Sardiu, Joshua M. Gilmore, Brad Groppe, Laurence Florens & Michael P. Washburn

Scientific Reports volume 7, Article number: 43845 (2017) (Web Link)

**[5]** Fu, L. and Fu, H. (2018) “Metric on the Countable Soft Topological Space”, Journal of Advances in Mathematics and Computer Science, 29(3), (Web Link)