Door space
In mathematics, in the field of topology, a topological space is said to be a door space if every subset is 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". Properties. Here are some facts about door spaces: To prove the first assertion, let formula_3 be a Hausdorff door space, and let formula_4 be distinct points. Since formula_3 is Hausdorff there are open neighborhoods formula_6 and formula_7 of formula_1 and formula_9 respectively such that formula_10 Suppose formula_9 is  an accumulation point. Then formula_12 is closed, since if it were open, then we could say that formula_13 is open, contradicting that formula_9 is an accumulation point. So we conclude that as formula_12 is closed, formula_16 is open and hence formula_17 is open, implying that formula_1 is not an accumulation point.