Definition of Door Space

Question

Every reference I can find regarding (topological) door spaces gives the following definition almost verbatim:

A door space is one in which every subset is either open or closed. [emphasis mine]

I can think of two interpretations of this definition. Which is the correct one?

1. A door space is one in which every subset is open, closed, or both.
2. A door space is one in which every nonempty proper subset is either open or closed, but not both.

The former seems plausible because no reference mentions the obvious caveat that the empty set and the entire set are clopen in every topology.

The latter seems plausible based on the fact that every definition uses the word "either" (which, to me, connotes exclusive or) and that physical doors cannot be clopen.

Answer 1

The only place where I could find a definition for "door space" is in Kelly's General Topology.

There, in Exercise C, page 76 (first edition), is written:

"A Hausdorff door space has at most one accumulation point, and if $x$ is a point which is not an accumulation point, then $\{x\}$ is open."

So, as singleton sets in Hausdorff spaces are closed, it seems Kelly is allowing the possibility that in a door space, a set may be both open and closed.

Answer 2

From John L. Kelley's General Topology (available at the Internet Archive), p. 76:

A topological space is a door space iff every subset is either open or closed.

I do not believe the exclusive-or interpretation is plausible, or that the word "either" is indicative of such an interpretation. In the same text, note the use of "either" in the definition of a $T_0$-space, p. 57:

A topological space is a $T_0$-space iff for each pair $x$ and $y$ of distinct points, there is a neighborhood of one point to which the other does not belong. In slightly different terminology, the space is a $T_0$-space iff for distinct points $x$ and $y$ either $x\notin\{y\}^-$ or $y\notin\{x\}^-$.