Limit points and the trivial topology: A textbook error? I'm reading George L. Cain's Introduction to General Topology, and am confused by the following example. We have:
Pg 32: Definition: Let $(X,T)$ be a topological space. If $S$ is a subset of $X$, a point $p$ in $X$ is a T-limit point of $S$ if every element of $T$ containing $p$ meets $S$ in a point other than $p$.
Pg 32: Example 2.4a: Let $X$ be the set of real numbers and let $T$ be the trivial topology ($ \emptyset, X$). Suppose $S$ is any nonempty subset of $X$. Then every $x \in X - S $ is a $T$-limit point of $S$.
It seems like every $x \in X$ should be T-limit point of $S$, unless  $S$ is a singleton, in which case the book would be correct. In the case where $S$  is not a singleton, I think I've even found a counter-example to the book. Consider $S=[0,2]$. Note that $p=1$ is a limit point of $S$ since all 1 elements of the topology containing $p=1$ meet $S=[0,2]$ at a point other than $p=1$. 
Who is correct about the T-limit points of S in Example 2.4a? If the book is correct, what is wrong with my counterexample?

You're both right- in your example, $1$ is indeed a limit point of $S$. But the text doesn't claim that those are the only $T$ limit points of $S$. It just doesn't want to go to deep into the detail of this examples just to say that if $S$ is a singleton, then all points of $X-S$ are $T$ limit points of $S$, and otherwise, all points in $S$ are $T$ limit points of $S$. It just gives you the general case which is true of any $S$, though you're free to investigate the particulars by yourself.
