On page 85 of the book it reads the following definition:
Definition 4. Suppose that for each pair of distinct points $x$ and $y$ in a topological space $T$, there is a neighborhood $O_x$ of $x$ and a neighborhood $O_y$ of $y$ such that $x \in O_y, y \in O_ x$. Then $T$ is said to satisfy the first axiom of separation, and is called a $T_1$- space.
But in other sources, such as Mathworld, they define
$T_1$-separation axiom : For any two points $x,y \in X$ there exists two open sets $U,V$ such that $x\in U$ and $y \notin U$ , and $y\in V$ and $x \notin V$.
Are both definitions equivalent?