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I'm not sure "at the level of Hartshorne" is so well-defined. A basic question about the definition of schemes might warrant closure, but a basic question on the theorem of formal functions sounds already quite different
When I've read the question it immediately translated in my mind as "Topology is for talking about converging sequences: what the heck is the Zariski topology about?". Probably I got a bit carried away in my willingness to reply that there's more to geometry than converging sequences. That said I think that the question may use a clarification
