empty subset

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

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

Last revised on January 9, 2017 at 08:32:28. See the history of this page for a list of all contributions to it.