Clarification in definition of neighborhood in Topology

Question

Suppose a topology $\tau$ is given on a set $X=\{a,b,c\}$.

Let $\tau=\{\emptyset, X,\{a\}\}$.

Let $a\in X$. Can I say that $\{a\}$ is a neighborhood of $a$?

Because of the fact that $a$ lies in $\{a\}$.

Kindly help.

Answer 1

Let $X$ be a topological space, and let $x \in X$.

$U \subseteq X$ is a neighborhood of $x$ if there exists an open set $A \subseteq X$ such that $x \in A \subseteq U$.

By the definition of neighborhood, an open set is a neighborhood of each of its points.

In your example, $\{a\}$ is open, so $\{a\}$ is a neighborhood of $a$ in your space.