Spaces not distinguishing pointwise and \(\mathcal {I}\)-quasinormal convergence. (Q2838739)

The authors introduce a new concept of \(\mathcal {I}\)-quasinormal convergence of functions, where \(\mathcal {I}\) is a fixed ideal of sets. This is a combination of two types of convergences: \(\mathcal {I}\)-convergence and equal (also called quasinormal) convergence. \(\mathcal {I}\)-quasinormal convergence is stronger than pointwise convergence.NEWLINENEWLINENew classes of topological spaces are introduced: \(\mathcal {I}QN\) and \(\mathcal {I}wQN\). A space is \(\mathcal {I}QN\) if pointwise convergence of continuous real functions coincides with the \(\mathcal {I}\)-quasinormal convergence. A weaker condition, involving \(\mathcal {I}\)-quasinormal convergent subsequences defines the class of \(\mathcal {I}wQN\) spaces.NEWLINENEWLINESome characterizations of \(\mathcal {I}\)-quasinormal convergence as well as some preservation properties of \(\mathcal {I}QN\) and \(\mathcal {I}wQN\) spaces are stated. Examples distinguishing several convergences involving an ideal are provided.NEWLINENEWLINEA remark concerning notation: `quasinormal convergence' is often called `equal convergence', an `ideal satisfying the Chain Condition' is just a `countably generated ideal' and an `AP-ideal' is the same as a `P-ideal'.