Intersection of Singleton

Theorem

Consider the set of sets $\mathbb S$ such that $\mathbb S$ consists of just one set $S$.

Then the intersection of $\mathbb S$ is $S$:

$\displaystyle \mathbb S = \left\{{S}\right\} \implies \bigcap \mathbb S = S$

Proof

Let $\mathbb S = \left\{{S}\right\}$.

Then from the definition:

$\displaystyle \bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$

from which it follows directly:

$\displaystyle \bigcap \mathbb S = \left\{{x: x \in S}\right\}$

as $S$ is the only set in $\mathbb S$.

That is:

$\displaystyle \bigcap \mathbb S = S$

$\blacksquare$

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{I}$