door space

A topological spaceMathworldPlanetmath X is called a door space if every subset of X is either open or closed.

From the definition, it is immediately clear that any discrete space is door.

To find more examples, let us look at the singletons of a door space X. For each xX, either {x} is open or closed. Call a point x in X open or closed according to whether {x} is open or closed. Let A be the collectionMathworldPlanetmath of open points in X. If A=X, then X is discrete. So suppose now that AX. We look at the special case when X-A={x}. It is now easy to see that the topologyMathworldPlanetmath τ generated by all the open singletons makes X a door space:


If BX does not contain x, it is the union of elements in A, and therefore open. If xB, then its complementPlanetmathPlanetmath Bc does not, so is open, and therefore B is closed. ∎

Since τ=P(A){X}, the space X not discrete. In additionPlanetmathPlanetmath, X and are the only clopen sets in X.

When X-A has more than one element, the situation is a little more complicated. We know that if X is door, then its topology 𝒯 is strictly finer then the topology τ generated by all the open singletons. McCartan has shown that 𝒯=τ𝒰 for some ultrafilterMathworldPlanetmathPlanetmath in X. In fact, McCartan showed 𝒯, as well as the previous two examples, are the only types of possible topologies on a set making it a door space.


  • 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
  • 2 S.D. McCartan, Door Spaces are identifiable, Proc. Roy. Irish Acad. Sect. A, 87 (1) 1987, pp. 13-16.
Title door space
Canonical name DoorSpace
Date of creation 2013-03-22 18:46:11
Last modified on 2013-03-22 18:46:11
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 54E99