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  "body": "\n## Summary\n\nAn interior algebra is a [[Boolean algebra|0XADJKM699]] with one more operation that behaves like the [[interior|0RZH6W98HK]] operation of a [[topological space|0CQHFPGJ45]].\nIt is the algebraic form of topology, in the same way as a Boolean algebra is the algebraic form of set operations.\nInterior algebras are the algebraic models of the [[modal logic|C3RZY20ETK]] S4, where the interior operation interprets \"necessarily\".\nThe dual notion, with a closure operation instead, is called a closure algebra.\n\n## Explanations\n\n### L1 human\n\nTake a topological space, such as the real line.\nFor any set of points you can take its interior, the points that have some room around them inside the set.\nThe interior of the interval $[0, 1]$ is $(0, 1)$.\nTaking the interior twice gives nothing new, and the interior of an intersection is the intersection of the interiors.\nAn interior algebra keeps only these rules and forgets the points.\n",
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