Definition of $T_1$ Space in Kolmogorov-Fomin Introductory Real Analysis book

Question

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?

Answer 1

It's certainly a typographical error. "$x\notin O_y$" and "$y\notin O_x$" was meant instead of "$x\in O_y$" and $y\in O_x$". (This is used in the Theorem immediately following the definition).

With this correction, the two notions coincide (trivially).

But to answer you directly, note that every topological space $T$ satisfies the criteria as written: just take $O_y=O_x=T$.

Answer 2

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.