Compactness Theory in Complex Neutrosophic Soft Spaces and Its Application to Visual Cosine Similarity Analysis

Abstract

In the framework of complex neutrosophic soft topology, the paper investigates locally compact spaces and complex neutrosophic soft compact spaces. Complex neutrosophic soft open covers, finite sub-covers, closed set intersections, centered families, and several comparable requirements are used to define the idea of compactness. Compact subsets of Hausdorff space were shown to be closed, and compact Hausdorff spaces were shown to be normal. The relationships between compactness axioms, closed ness axioms, Hausdorff ness axioms, normalcy axioms, and separation axioms (\(T_1\), \(T_2\), \(T_3\)) are investigated. The concept of complicated neutrosophic soft local compactness, which is a weakening of compactness expressed in terms of neighborhoods with compact closures, is then defined. Refined neighborhood confinement is established, and it is shown that both open and closed subspaces of locally compact Hausdorff spaces exhibit local compactness. These results provide a solid foundation for modeling the localized uncertainty and are generalized to the complex neutrosophic soft environment from the classical compactness theory. Using both quantitative and advanced visualization methods, this research offers a thorough and graphical analysis of the cosine similarity trends between the different dimensional cues and class templates. The strongest relationship is always found between Signal \(S_2\) and Template \(T_1\). Each of the cosine similarity scores is evaluated separately to display the predominant, moderate, and weak signal-template cross section. Through a number of complementary visual techniques such as normalized heatmaps, 2D and 3D bar graphs, surface plots, spline interpolations, PCA projection, and t-SNE embedding, the similarity matrix is rendered as an informative geometric structure in the form of peaks, ridges, and valleys. These visualizations offer an insightful understanding of the level of agreement, robustness and sensitivity by exhibiting stable similarity points, transitions and fragile points. Moreover, stable and sensitive matches where the confidence for classification is the highest and where we need to be careful are detected through the sensitivity and derivative analysis of the landscape. In general, the hybrid visual-metric approach provides a formal yet intuitive way to understand the correlation between signal and template that explains why the cosine similarity is a successful method of pattern recognition, clustering, and interpretation of multidimensional data.