Effective local connectivity properties

TL;DR

This paper explores effective versions of local connectivity properties in Euclidean spaces, establishing equivalences and demonstrating that certain computably compact continua are also computably arcwise connected.

Contribution

It introduces effective analogs of local connectivity and proves their equivalence, showing that computably compact, effectively locally connected continua are also computably arcwise connected.

Findings

Effective local connectivity is equivalent to uniform local arcwise connectivity.

Computably compact, effectively locally connected continua are also computably arcwise connected.

Abstract

We investigate, and prove equivalent, effective versions of local connectivity and uniformly local arcwise connectivity for connected and computably compact subspaces of Euclidean space. We also prove that Euclidean continua that are computably compact and effectively locally connected are computably arcwise connected.