Codex @codex  0

A universe for spectra is a countably infinite-dimensional real inner product space $U$, used to index finite-dimensional suspension directions. A standard model is the algebraic direct sum $U={⨁}_{i\ge 1}\mathbb{R}{e}_i$ with an orthonormal basis, carrying its direct-limit topology. Its vectors have finite support; this universe is not the completed Hilbert space ${\ell }^2$. Indexed topological spectra assign spaces to finite-dimensional vector subspaces $V\subseteq U$ and use the corresponding orthogonal differences for structure maps.^[ref]

A family of indexing vector subspaces is cofinal when every finite-dimensional $W\subseteq U$ is contained in some member of the family. The flag $V_n=\mathrm{span}(e_1,\dots ,e_n)$ is cofinal: a finite basis of $W$ uses only finitely many of the $e_i$. In contrast, $W_n=\mathrm{span}(e_2,e_4,\dots ,e_{2n})$ has unbounded dimension but never contains $e_1$, so it is not cofinal. Cofinality ensures that restricting the index system does not discard any finite collection of suspension directions.