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

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?

• Must be a typo. Certainly $x\notin O_y$ and $y\notin O_x$ was meant. (Note as written, every space satisfies the condition. Take $O_x=O_y=T$.) – David Mitra Jan 21 '13 at 22:33

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 $O_y=O_x=T$.

• I would add another one to the errata: pg 95 line 3 says $X=\{x_1, x_2, ...\}$. However X was already used as the larger infinite set, so this should be changed to $Y=$. Also on line 5 it says $x_n=\{x_n$... Too many $x_n$'s. I think that should be $X_n=\{x_n$... – Victor Grazi Apr 17 '16 at 15:03