Estimation of the number of one-point expansions of a topology which is given on a finite set (Q2883526)

A one-point expansion of a topology \(\tau\) on a finite set \(X\) is a topology \(\tau'\) on \(X\cup \{y\}\) (with \(y\not\in X\)) whose restriction to \(X\) is \(\tau\). If \(m\) is the number of open sets of \(\tau\), the author shows that the number \(t(\tau)\) of one-point expansions of \(\tau\) satisfies the inequalities NEWLINE\[NEWLINE2m+\log_2m-1~ \leq~ t(\tau)~ \leq~\frac{m(m+3)}{2} -1NEWLINE\]NEWLINE Moreover, the upper bound is reached if and only if \((X,\tau)\) is a linearly ordered set and the lower bound is reached if and only if \(\tau\) is a an atomistic lattice (a lattice is said \textsl{atomistic} if every non minimal element is the join of atoms). This result is obtained by exploiting an algorithm for constructing one-point expansions developed in a previous paper of the author and \textit{A. V. Kochina} [Bul. Acad. Stiinte Repub. Mold., Mat. 2010, No. 3(64), 67--76 (2010; Zbl 1217.54003)].