Preimage Cardinalities of Continuous Functions

TL;DR

This paper characterizes all subsets of natural numbers that can serve as the set of preimage cardinalities for continuous functions, exploring various subclasses of these functions.

Contribution

It provides a complete classification of possible preimage cardinality sets for continuous functions and their subclasses.

Findings

Identifies all subsets of natural numbers realizable as preimage cardinalities.

Analyzes preimage cardinalities for different subclasses of continuous functions.

Provides a comprehensive framework for understanding preimage structures.

Abstract

We find all subsets of $\mathbb{N}$ which occur as the set of possible cardinalities of preimages of a continuous function. We also study and answer this question for various subclasses of continuous functions.