Definition of Tychonoff Separation Property in Royden's Real Analysis

Question

In the fourth edition of Royden's Real Analysis (Section 11.2) he defines the Tychonoff Separation Property by

Tychonoff Separation Property For each two points $u$ and $v$ in $X$, there is a neighborhood of $u$ that does not contain $v$ and a neighborhood of $v$ that does not contain $u$.

However this seems to actually be the definition of the $T_1$ separation property according to this Wikipedia article as well as other texts such as Knapp's Basic Real Analysis. Am I mistaken about the discrepancy or does Royden just define Tychonoff separation differently?

Answer 1

You are right. I would not like to use the phrase "Tychonoff Separation Property" as a synonym for $T_1$, but we must be aware that mathematical notation is not standardized. Sometimes it even has a geographic or cultural background. For example, there is a topological space known as the "Moore plane" in the United States and as the "Niemytzki plane" in Eastern Europe.

Concerning notational variants of separation axioms see https://en.wikipedia.org/wiki/Separation_axiom.