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Door not implies discrete

This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., Door space (?)) need not satisfy the second topological space property (i.e., Discrete space (?))
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It is possible for a topological space to be a door space (i.e., every subset is either an open subset or a closed subset) but not a discrete space.


Further information: Sierpiński space

Consider the Sierpiński space: a set X with two distinct elements a,b. Define the open subsets of X as the subsets:

\{ \}, \{ a \}, \{ a,b \}

Correspondingly, the closed subsets are:

\{ \}, \{ b \}, \{ a,b \}

This is a door space, because every subset is either open or closed (and some are both). On the other hand, it is not discrete because not every subset is open.