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This paper introduces the trivial fiber topology on schemes. Forone-dimensional base schemes, we use it to describe fibrant replacements in thestable motivic homotopy category and motivic infinite loop spaces. We alsoextend the Garkusha-Panin and Voevodsky strict $\mathbb{A}^{1}$-invariancetheorems to one-dimensional base schemes. The trivial fiber topology plays acentral role in the proof of refined localization results for motivic homotopycategories. Moreover, we extend Morel's $\mathbb{A}^{1}$-connectivity theoremon Nisnevich sheaves of stable motivic homotopy groups. These results open newvistas for computations of motivic invariants over deeper base schemes ofarithmetic interest.

The trivial fiber topology and framed motives over the integers