The \'{e}tale sectional number is either 1 or infinity

Abstract

In this work, we show that the \'{e}tale sectional number (\text{\'{E}tale-sec$(-)$}), i.e., the sectional number in the category of topological spaces with the \'{e}tale quasi Grothendieck topology (as defined in arXiv:2410.22515), is either 1 or infinity. Specifically, given a continuous map $f:X\to Y$, we demonstrate that \[\text{\'{E}tale-sec$(f)$}=\begin{cases} 1,&\hbox{ if $f$ is locally sectionable,} \infty,&\hbox{ if $f$ is not locally sectionable.} \end{cases} \] Additionally, for a path-connected space $X$, the \'{e}tale topological complexity satisfies \[\text{TC}_{\text{\'{e}tale}}(X)=\begin{cases} 1,&\hbox{ if $X$ is locally contractible,} \infty,&\hbox{ if $X$ is not locally contractible.} \end{cases} \] These results provide a way to understand the \aspas{complexity} of maps and spaces within the context of the \'{e}tale quasi Grothendieck topology, a structure that considers local behavior of maps and spaces. The classification into values of 1 or infinity reflects a dichotomy in the local geometric structure of the map or space, with the presence or absence of local sections or contractibility significantly influencing the outcome.