The étale sectional number is either 1 or infinity

In this work, we show that the étale sectional number ($\acute{\text{E}}\text{tale-sec}(-)$), i.e., the sectional number in the category of topological spaces with the étale 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 $$\acute{\text{E}}\text{tale-sec}(f)=\{\begin{array}{lll}1,& \text{ if }f\text{ is locally sectionable,}\mathrm{\infty },& \text{ if }f\text{ is not locally sectionable.}\end{array}$$ Additionally, for a path-connected space $X$, the étale topological complexity satisfies $${\text{TC}}_{\acute{\text{e}}\text{tale}}(X)=\{\begin{array}{lll}1,& \text{ if }X\text{ is locally contractible,}\mathrm{\infty },& \text{ if }X\text{ is not locally contractible.}\end{array}$$ These results provide a way to understand the \aspas{complexity} of maps and spaces within the context of the étale 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.