Shy Maps in Topology

TL;DR

This paper introduces the concept of shy maps in topology, defining them as continuous functions where inverse images of path-connected sets are path-connected, and explores their properties and implications for fundamental groups.

Contribution

It extends the concept of shy maps from digital topology to general topological spaces and analyzes their basic properties and algebraic implications.

Findings

Shy maps induce surjections on fundamental groups when the target space is semilocally simply connected.

Not all shy maps onto non-semilocally simply connected spaces induce fundamental group surjections.

Basic properties of shy maps in topological spaces are established.

Abstract

There is a concept in digital topology of a shy map. We define an analogous concept for topological spaces: We say a function is shy if it is continuous and the inverse image of every path-connected subset of its image is path-connected. Some basic properties of such maps are presented. For example, every shy map onto a semilocally simply connected space induces a surjection of fundamental groups (but a shy map onto a space that is not semilocally simply connected need not do so).