ℵ 0 ‐categorical Banach spaces contain ℓp or c 0

This paper has three parts. First, we establish some of the basic model theoretic facts about $M_{\mathcal{T}}$, Tsirelson space Figiel and Johnson \cite{FJ}. Second, using results first part, give general Banach spaces. Third, study model-theoretic dividing lines in spaces their theories. In~particular, show: (1) $M_{\mathcal{T}}$ \emph{non independence property} (NIP); (2) every that is $\aleph_0$-categorical up to small perturbations embeds $c_0$ or $\ell_p$ ($1\leqslant p<\infty$) almost isometrically; consequently (continuous) first-order theory does not characterize isometric isomorphism.