I'm trying to learn a bit about topology through independent study. I've been using Bert Mendelson's "Introduction to Topology - 3rd edition". I'm having a lot of fun but I'm a bit confused regarding definition 4.9 on pg 45. I will reproduce it hereafter:
Definition 4.9 - Let $a$ be a point in metric space $X$. A collection of neighborhoods $\mathcal{B}_a$ is called a basis for the neighborhood system at $a$ if every neighborhood $N$ of $a$ contains some element $B$ in $\mathcal{B}_a$.
Here is the source of my confusion, please correct me if I am wrong:
1) Every neighborhood of $a$ must contain $a$ it self, so shouldn’t any neighborhood of $a$ be automatically a basis of the neighborhood system at $a$?
2) If this is true (and I'm hoping it is false) what is the condition that forces $\mathcal{B}_a$ to grow beyond a trivial set such as $\mathcal{B}_a = \{a\}$?
Thank you.