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 (or both). 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: A Hausdorff door space has at most one accumulation point. In a Hausdorff door space if x is not an accumulation point then {x} is open. To prove the first assertion, let X be a Hausdorff door space, and let x ≠ y be distinct points. Since X is Hausdorff there are open neighborhoods...