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In this paper, we are going to analyze a wide class of topological spaces - the so-called metric spaces. These are sets in which is defined the notion of distance between any two points. It is well known that the distance function or metric defined on a metric space X induces a topology on that space X. In the second part of the paper, we are going to study well-known characterizations of the class of topological spaces, the topology of which is determined with the help of metrics - the so-called metrizable spaces. The question here is when is a topological space metrizable? The answer of this question is the main result of this research, which are some important criteria, who are necessary and sufficient, that topological spaces must possess in order to be metrizable. [ABSTRACT FROM AUTHOR]

In this paper, we are going to analyze the relations between different types of convergence of a random sequence, such as almost sure convergence, convergence in mean square, convergence in distribution and convergence in probability. The convergence in distributions says nothing about the relationship between the random variables Xn and X and X, while for convergence in probability, the joint distribution of Xn and X is relevant. In the main part of the paper, we are going to prove the theorem which argues that the convergence in probability implies convergence in distribution, and the opposite is not true. But if ..., where c is a constant, then ... which mean that convergence in probability to a constant is equivalent to convergence in distributions. Also, we give some interesting examples. [ABSTRACT FROM AUTHOR]