I'm reading Gathmanns's lecture notes on Algebraic Geometry where he defines a Prevariety as ringed space $V$ which has a finite open cover of affine varieties $U_i$. I assume this not just a set-theoretic covering, and that the topologies/structure sheaves of the $U_i$ has to play nice with the topology/structure sheaf of $V$, but I can't find a definition which specifies this anywhere. For example, it seems sensible to require that the subspace topologies $V \cap U_i$ agree with the original topologies of the $U_i$ and that $\mathcal{O}_V(U_i) = \mathcal{O}_{U_i}(U_i)$. Is there such compatibility condition, and in that case, what is it?
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A prevariety is a ringed space $(X, \mathscr{O}_X)$ such that there exists a finite open covering $U_1, \ldots, U_n$ of $X$, where each ringed space $(U_i, \mathscr{O}_{U_i})$ is an affine variety. This means there are affine varieties $V_i \in \mathbb{A}^{n_i}$ and ringed space isomorphisms $\psi_i \colon (U_i, \mathscr{O}_{U_i}) \to (V_i, \mathscr{O}_{V_i})$.
Here, the topology of $U_i$ is the subspace topology, and $\mathscr{O}_{U_i}$ is just the restriction of $\mathscr{O}_X$ to $U_i$.
