The purpose of this paper is to define new classes of analytic functions by amalgamating the concepts of q-calculus, Janowski type functions and $(x,y)$-symmetrical functions. We use the technique of convolution and quantum calculus to investigate the convolution conditions which will be used as a supporting result for further investigation in our work, we deduce the sufficient conditions, P$\acute{o}$lya-Schoenberg theorem and the application. Finally motivated by definition of the neighborhood, we give analogous definition of neighborhood for the classes ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$, and then investigate the related neighborhood results, which are also pointed out.

A topological space X is called C-Tychonoff if there exist a one-to-one function f from X onto a Tychonoff space Y such that f restriction K from K onto f(K) is a homeomorphism for each compact subspace K of X. We discuss this property and illustrate the relationships between C-Tychonoffness and some other properties like submetrizability, local compactness, L-Tychononess, C-normality, C-regularity, epinormality, sigma-compactness, pseudocompactness and zero-dimensional.

We apply some topological concepts on topological spaces generated from both equality and similarity relation for our information system, which examine 10 hypothetical patients who suffer from some COVID−19 symptoms. This is based on the degree of accuracy generated from the cardinality of the lower and upper approximation. This method is clarified by application.