Is the 'Locally Ringed' condition in the definition of a Scheme redundant? My question is, if it admits a cover by Affine Schemes, does it follow that the Ringed space is Locally Ringed?
More generally, if a Ringed Space $(X,\mathscr{O}_X)$ admits a cover $\{U_i\}$ such that each $(U_i,{\mathscr{O}_{X}}_{|U_i})$ is Locally Ringed, is $(X,\mathscr{O}_X)$ itself Locally Ringed?