Cauchy nets and convergent nets on semilinear topological spaces (Q444719)

The author finds some new differences between semilinear topological spaces (s.t.s.) and linear topological spaces by some Cauchy-type conditions for nets. Starting from the idea that the definition of a Cauchy net in linear topological spaces can be equivalently formulated by one of the two conditions: for every \(V\in{\mathcal V}(0)\) there exists \(i_V\in I\) such that for all \(i,j\geq i_V\) we have \(x_i- x_j\in V\), or \(x_i\in x_j+ V\), the author shows that in s.t.s. nets that satisfy the above conditions, called Cauchy nets in difference and translated Cauchy nets, respectively, can be connected to two specific types of convergence. In addition, a third Cauchy-type net called a semi-Cauchy net is also defined.NEWLINENEWLINE Furthermore, the author also characterizes Cauchy nets using small sets, establishes some relationships with the above concepts of convergence by Cantor-type theorems, and compares them.