Some metricability theorems

In this thesis, studies on the metricability problem, which is one of the main study subjects of topology, expressed with the question "Which topological properties belonging to a (X,τ) topological space can make the τ topology be equal to the τ_d topology determined by an appropriate metric d on X ?" has been researched and compiled. In this compilation thesis study, the necessary and sufficient conditions for the metrization of a topological space X as expressed by the Nagata-Smirnov Metricability Theorem, are that "X space must have a regular and 𝜎-local finite base" and as expressed by the Smirnov Metricability Theorem, "local metricability of X space is a paracompact Hausdorff space."