empty subset

Given a set AA, the empty subset of AA, denoted A\empty_A, is the subset of AA defined by the property that, for every element xx of AA, it is false that xx belongs to A\empty_A.

The underlying set (or shadow) of any empty subset is the empty set. That is, if we interpret A\empty_A as an injective function SAS \hookrightarrow A, then the source SS of this function is the empty set.

In the usual framework of material set theory, every empty subset is identical to the empty set. For this reason, it is common to write simply \empty instead of A\empty_A. Even from a structural perspective, this is an abuse of language that is unlikely to cause any confusion.

In the context of topology, we often speak of the empty subspace. In point-set topology, this is indeed an empty subset of the set of points, but in point-free topology, a space is not necessarily the empty space just because it has no points, and the empty subspace is similarly subtle.

Revised on January 9, 2017 08:32:28 by Toby Bartels (