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  "body": "\n## Summary\n\nA rigid analytic space is the analogue of a complex analytic space over a complete non-archimedean field such as the [[$p$-adic numbers|BT6T702C28]].\nIt is glued from affinoid spaces, the maximal spectra of quotients of algebras of convergent power series, along a Grothendieck topology of admissible opens.\nJohn Tate introduced the theory to describe elliptic curves with split multiplicative reduction as quotients of the multiplicative group.\nIt underlies $p$-adic uniformization and $p$-adic Hodge theory.\n\n## Explanations\n\n### L1 human\n\nOver the $p$-adic numbers, the usual topology is totally disconnected: every disc splits into smaller disjoint open discs.\nWith this topology, functions defined piece by piece behave badly, and there is no good notion of analytic continuation.\nRigid analytic geometry fixes this by allowing only certain coverings, the admissible ones.\nThe closed unit disc, for instance, is not covered admissibly by the open unit disc and the circle $\\abs{x} = 1$.\nWith this restriction, analytic functions on $p$-adic spaces behave much like holomorphic functions on complex spaces.\n",
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