Countable open and closed functions

TL;DR

This paper introduces 2-open and 2-closed functions as natural variants of open and closed functions, demonstrating their properties through the construction of specific subsets that cover the target space.

Contribution

It defines new classes of functions, 2-open and 2-closed, and explores their properties, bridging the gap between open/closed functions and their countable counterparts.

Findings

Existence of subsets where the functions are open or closed onto their images

Coverage of the entire target space by these subsets

New classes of functions with properties similar to open and closed functions

Abstract

We define two natural classes of functions, called 2-open and 2-closed, that are closest to open and closed functions. We show that they have the following property: there are $X_i \subset X$ $ (i=1,2,...$) such that $f|X_i$ are open or closed functions onto $f(X_i)$ and $f(X_i)$ cover $Y.$