Subset

A subset of a set $B$ is a set $A$ such that every element of $A$ is an element of $B$. Formally,

Subset inclusion $\subseteq$ is the basic relation for comparing sets. Many constructions in analysis (closures, neighborhoods, function spaces) are naturally expressed as subsets of larger ambient sets.

Examples:

• $\{1,2\}\subseteq \{1,2,3\}$.
• $\mathbb{Q}\subseteq \mathbb{R}$.
• For every set $A$, $\varnothing \subseteq A$.