A Cartesian closed topological category of sequential spaces (Q2644753)

The category of sequential spaces SEQ is introduced. Let X be a set and \(\pi\) a mapping which to every point \(x\in X\) assigns a set \(\pi\) (x) of sequences of points of X. The pair (X,\(\pi\)) is called a sequential space provided the following two axioms are satisfied: (i) If \(x\in X\) and \(x_ i=x\) for all \(i\in {\mathbb{N}}\), then \(\{x_ i\}_{i\in {\mathbb{N}}}\in \pi (x).\) (ii) If \(\{\{x_{ij}\}_{j\in {\mathbb{N}}^ 0}\}_{i\in {\mathbb{N}}^ 0}\) is a sequence of sequences of points of X such that \(\{x_{ij}\}_{j\in {\mathbb{N}}}\in \pi (x_{i0})\) for all \(i\in {\mathbb{N}}^ 0\) and \(\{x_{ij}\}_{i\in {\mathbb{N}}}\in \pi (x_{0j})\) for all \(j\in {\mathbb{N}}^ 0\), then \(\{x_{ii}\}_{i\in {\mathbb{N}}}\in \pi (x_{00}).\) SEQ is the category of all sequential spaces with morphisms defined by: Given two sequential spaces \(A=(X,\pi)\) and \(B=(Y,\rho)\) then a mapping \(f:X\to Y\) is a morphism from A to B provided that \(\{x_ i\}_{i\in {\mathbb{N}}}\in \pi (x)\) implies \(\{f(x_ i)\}_{i\in {\mathbb{N}}}\in \rho (f(x))\) for every \(x\in X.\) It is proved that the category SEQ is topological and cartesian closed. Finally, subcategories of the category SEQ are investigated.