TITLE: Definition of $T_1$ Space in Kolmogorov-Fomin Introductory Real Analysis book
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: Google

Kolmogorov Fomin errata

and find my errata sheets still out there on the internet.  (Yes, this one is listed.)  K & F is a good book for an introduction to the topics, plus it has a great price.