I've found these two different definitions of a neighborhood basis, and I was wondering if the definitions are equivalent to one another.
In the following, let $(X,\tau)$ be a topological space.
Def 1: For $x\in X$, let $\mathcal{B}_x$ be a collection of neighborhoods of $x$. We say $\mathcal{B}_x$ is a neighborhood basis at $x$ if and only if for each neighborhood $U$ of $x$, there is an $B_x\in\mathcal{B}_x$ so that $B_x\subseteq U$.
Def 2: For $x\in X$, let $\mathcal{B}_x$ be a collection of neighborhoods of $x$. We say $\mathcal{B}_x$ is a neighborhood basis at $x$ exactly when the following condition is satisfied:
$$U\in\tau \iff \forall x\in U, \exists B_x\in\mathcal{B}_x :x\in B_x\subseteq U.$$
Best Answer
The definitions are not equivalent, but just because they talk about objects of “different types”. In the first definition we have a fixed point $x ∈ X$ and one collection $\mathcal{B}_x$ whose property we are defining. In the second definition it seems that we are in the same situation, but the defining equivalence talks about different collections $\mathcal{B}_x$ at different points. So the second definition really should be
Def 2: Let $\{\mathcal{B}_x\}_{x ∈ X}$ be a system of collections of neighborhoods at the respective points. We say that $\{\mathcal{B}_x\}_{x ∈ X}$ is a neighborhood basis system if \begin{align} U ∈ τ \iff ∀x ∈ U,\ ∃B_x ∈ \mathcal{B}_x: x ∈ B_x ⊆ U. \end{align}
Now you can ask if a collection $\{\mathcal{B}_x\}_{x ∈ X}$ is a system of neighborhoods bases at the respective points (according to Def 1) if and only if it is a neighborhood basis system (according to Def 2). The answer is yes, and since we clarified the type issues, you may try to prove it. Just note that when going from Def 1 to Def 2, you want to prove the equivalence in Def 2, so you prove “Def 1 + LHS of Def 2 → RHS of Def 2” and “Def 1 + RHS of Def 2 → LHS of Def 2”. Then you also want to prove “Def 2 → Def 1”.