I was reading through Loring Tu's An Introduction to Manifolds, and I came across this example in the appendix:
For any set $S$, let $\tau$ be the collection of all subsets of $S$. Then $\tau$ is a topology on $S$, called the discrete topology. A singleton set is a set with a single element. The discrete topology can also be characterized as the topology in which every singleton subset $\{p\}$ is open. A topological space having the discrete topology is called a discrete space. The discrete topology is the finest topology on a set.
Right before this example, the author provides a lemma called local criterion for openness, but I am unable to see why a singleton in this situation is open.
EDIT:
Local Criterion for Openness:
Lemma A.2 (Local criterion for openness). Let $S$ be a topological space. A subset $A$ is open in $S$ if and only if for every $p ∈ A$, there is an open set $V$ such that $p ∈ V ⊂ A$.
Proof.
($⇒$) If $A$ is open, we can take $V = A$.
($⇐$) Suppose for every $p ∈ A$ there is an open set $V_p$ such that $p ∈ V_p ⊂ A$. Then $$ A \subset \bigcup_{p \in A} V_p \subset A $$ so that equality $A = \bigcup_{p \in A} V_p$ holds. As a union of open sets, $A$ is open.