nLab empty subset

Idea

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.

Related concepts

• empty set, empty subset

• empty type, empty context

• empty family, empty function, empty bundle

• empty topological space, empty simplicial set

• empty graph, empty category, empty diagram

• nothing