  This paper introduces the trivial fiber topology on schemes. For
one-dimensional base schemes, we use it to describe fibrant replacements in the
stable motivic homotopy category and motivic infinite loop spaces. We also
extend the Garkusha-Panin and Voevodsky strict $\mathbb{A}^{1}$-invariance
theorems to one-dimensional base schemes. The trivial fiber topology plays a
central role in the proof of refined localization results for motivic homotopy
categories. Moreover, we extend Morel's $\mathbb{A}^{1}$-connectivity theorem
on Nisnevich sheaves of stable motivic homotopy groups. These results open new
vistas for computations of motivic invariants over deeper base schemes of
arithmetic interest.
