Apparent contradiction Between Theorem About Hausdorff Topologies and Discrete Sets I am currently studying John M. Lee's "Topological Manifolds", where in chapter 2 it states the following:

Suppose $X$ is a Hausdorff space and $A\subseteq X$. If $p\in X$ is a limit
  point of $A$, then every neighborhood of $p$ contains infinitely many points of $A.$

However, just previous to this, the book also stated that discrete sets have a Hausdorff space (the discrete topology). It appears to me that these two theorems are in contradiction, so what am I missing?

You probably are missing a detail in the definition of a limit point,
Let $S \subset X$, where $X$ is a topological space. A point $x \in X$ is a limit point of $S$, if for each neighbourhood $V$ of $x$, $(V \setminus\{x\}) \cap S \neq \varnothing$.
Equivalently, $x$ is a limit point of $S$ if and only if $x \in \overline{S\setminus\{x\}}$.
From that follows that in a discrete space, no set has a limit point, since each point $x$ has a neighbourhood $V$ such that $V\setminus\{x\} = \varnothing$.
A subspace $D \subset X$ that is discrete in the subspace topology may have limit points in $X$, though (example: $\{ \frac1n : n \in \mathbb{Z}^+\} \subset \mathbb{R}$ has $0$ as a limit point, but is discrete in the subspace topology). But none of the limit points lies in $D$.
