Door space

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In mathematics, in the field of topology, a topological space is said to be a door space if every subset is either open or closed. The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither".

Here are some easy facts about door spaces:

• The discrete space is a door space.
• A Hausdorff door space has at most one accumulation point.

To prove the second assertion, let X be a Hausdorff door space. Suppose for contradiction that there are two distinct accumulation points x and y. Since we are working in a door space it will be important to remember that any subset of X is open or closed. If {x} (similarly for {y}) is open then we have a problem in that since x is a limit point and {x} is an open neighborhood of x then by the definition, ∅≠{x}\{x}∩X, but clearly {x}\{x}∩X=∅. Since X is Hausdorff there are open neighborhoods U and V of x and y respectively such that U∩V=∅. We argue that U\{x}∪{y} is closed. If it were open, then we could say that {y}=(U\{x}∪{y})∩V is open. So we conclude that as U\{x}∪{y} is closed, X\(U\{x}∪{y}) is open and hence {x}=U∩[X\(U\{x}∪{y})] is open and hence we have a contradiction.

• In a Hausdorff door space if x is not an accumulation point then {x} is open.

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