# Door not implies discrete

From Topospaces

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 (?)) neednotsatisfy the second topological space property (i.e., Discrete space (?))

View a complete list of topological space property non-implications | View a complete list of topological space property implications |Get help on looking up topological space property implications/non-implications

Get more facts about door space|Get more facts about discrete space

## Statement

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.

## Proof

`Further information: Sierpiński space`

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

Correspondingly, the closed subsets are:

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.