On disconnected images of connected spaces

TL;DR

This paper explores the concept of Boolean images of connected compact spaces, extending the idea from zero-dimensional spaces to connected ones, and investigates the properties and questions related to these images.

Contribution

It introduces the notion of Boolean images for connected compact spaces, expanding the framework beyond zero-dimensional cases and addressing new questions in the topology of compacta.

Findings

Defined Boolean images for connected compacta

Identified key properties distinguishing these images

Raised open questions about their structure and characteristics

Abstract

We introduce the notion that a zero-dimensional compact space $L$ is a \emph{Boolean image} of an arbitrary compact space $K$. When $K$ is also zero-dimensional, this just means that $L$ is a continuous image of $K$. However, a number of interesting questions arise when we consider connected compacta $K$.