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  "body": "\n## Summary\n\nPointless topology studies spaces through their lattices of open sets instead of through their points.\nIts basic objects are frames, complete lattices in which finite meets distribute over arbitrary joins, and locales, which are frames viewed as generalized spaces.\nEvery topological space gives a locale, its frame of open sets, and for [[sober spaces|4N4DNDR22Z]] no information is lost.\nSome theorems that need the axiom of choice for spaces, such as the product theorem for compact spaces, hold for locales without it.\n\n## Explanations\n\n### L1 human\n\nIn ordinary topology a space is a set of points together with a choice of open sets.\nPointless topology forgets the points and keeps only the open sets, with their order by inclusion, their unions and their finite intersections.\nThe open sets of the real line, for example, form such a structure, and the points can be recovered from it.\nWorking without points makes some constructions simpler and more constructive.\nIt also allows \"spaces\" that have open sets but no points at all.\n",
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