# Definition of a Prevariety - Specifically a covering by ringed spaces

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?

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 $$U_i$$.