Door space

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).^[1] 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 U and V of x and y respectively such that U ∩ V = ∅. Suppose y is an accumulation point. Then U \ {x} ∪ {y} is closed, since if it were open, then we could say that {y} = (U \ {x} ∪ {y}) ∩ V is open, contradicting that y is an accumulation point. 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, implying that x is not an accumulation point.

Notes

↑ Kelley, ch.2, Exercise C, p. 76.

References

Kelley, John L. (1991). General Topology. Springer. ISBN 3540901256.

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