On the structure of the matrix corresponding to a finite topology (Q1340390)

Let \(X = \{x_ 1, x_ 2, \dots, x_ n\}\) be a finite set and \(T\) a topology on \(X\). Let \(U_ i\) be the smallest open set containing \(x_ i\). Then define an \(n \times n\) zero-one matrix \(K = [k_{ij}]\) by \[ k_{ij} = \begin{cases} 1 & \text{if \(x_ j \in U_ i\)}\\0 & \text{if \(x_ j \notin U_ i\).}\end{cases} \] Let \(K\) be an \(n \times n\) matrix corresponding to a topology on an \(n\)-element set, denote by \(L(K)\) the set of all \((n + 1) \times (n + 1)\) matrices of the form \(\left[ \begin{smallmatrix} 1\;A\\ B\;K\end{smallmatrix} \right]\) corresponding to a topology on an \((n+1)\)-element set, where \(A\) is a \(1 \times n\) matrix and \(B\) is an \(n \times 1\) matrix. Then \(\alpha(K)\) is the number of elements of the set \(L(K)\). The paper studies numbers \(\alpha(K)\) for a matrix \(K\) which corresponds to a \(T_ 0\)-topology. It is proved that for an arbitrary matrix \(K\) of order \(n\) which corresponds to a \(T_ 0\)-topology \[ n(n+5)/2 + 1 \leq \alpha(K) \leq 2^{n + 1} + n - 1. \] The proof is based on the following two results (also proved in the paper): (1) Denote by \(I_ n\) the identity matrix, then \(\alpha(K) \leq \alpha(I_ n)\) for every matrix \(K\) corresponding to a topology. (2) Denote by \(L_ n\) the upper triangular matrix \(L_ n = (\ell_{ij})\), where \(\ell_{ij} = 1\) iff \(i \leq j\). Then \(\alpha(L_ n) \leq \alpha(K)\) for every matrix \(K\) corresponding to a \(T_ 0\)- topology.