# How to say that one set is the smallest containing another set?

Imagine that we have a collection of sets $$\cal C$$, let $$S\notin \cal C$$ and $$A \in \cal C$$.

In algebra, topology, etc... we often see statements like " ----- is the smallest set containing -----", but I'm always not sure how to formalize these kinds of statements. How can I write that $$A$$ is the smallest set in $$\cal C$$ containing the set $$S$$? How can we measure the "length" of these sets in order to say that one is smaller than the other?

• For sets of discrete elements $|S|$ is the cardinality or "size" of a set—the number of its elements. Nov 9, 2020 at 16:10
• But, for example, If I'm talking about sets in $\Bbb R$, how can I say that some set is the smallest set containing another? All intervals in $\Bbb R$ have the same cardinality @DavidG.Stork Nov 9, 2020 at 16:11

In this use we order sets by inclusion, so you would say that $$A$$ is in $$\mathcal C$$, contains $$S$$, and all other sets in $$\mathcal C$$ that contain $$S$$ also contain $$A$$.

• So, $S \subset B \Rightarrow A \subset B$? Nov 9, 2020 at 16:12
• Yes, for all sets $B \subset \mathcal C$ Nov 9, 2020 at 16:19

The statement usually means the property $$A\in\mathcal C\land A\supseteq S\land (\forall B\in \mathcal C,(A\subseteq B\lor S\nsubseteq B))$$ id est that $$A\in\mathcal C$$ is a set containing $$S$$ and, for all other sets $$B\in\mathcal C$$ which contain $$S$$, $$A\subseteq B$$.

Personally, I would not use the word "smallest" when I'm referring to cardinality. I would say "of least cardinality" (or "least size", or "least measure", or "least greatest lower bund of the sine of an odd integer contained in it"...).

• So is this the same as saying that if $\mathcal {P} = \{U \in \mathcal C : S \subseteq U\}$, then $\bigcap_{U \in \mathcal P} U = A$? Nov 9, 2020 at 16:16
• @EduardoMagalhães There is one if and only if $\bigcap\mathcal P\in\mathcal C$, in which case $A=\bigcap \mathcal P$.
– user239203
Nov 9, 2020 at 16:18