Method for constructing one-point expansions of a topology on a finite set and its applications

The article consists of two parts. In the first part we present an algorithm which allows to receive, for any topology $\tau$ which is given on a set $X $ from $n$ elements, all topologies on the set $X\bigcup\{y\}$ each of which induces the topology $\tau$ on the set $X $. In the second part (as an example) this algorithm is applied for calculation of the number of topologies on the set $Y$ each of which induces the discrete topology on the set $X$.