The definition my lecturer gave me for a limit point in a metric space is the following:
Let $(X, d)$ be a metric space and let $Y \subseteq X$. We say that a point $x \in X$ is a limit point of $Y$ if for any open neighborhood $U$ of $x$ the intersection $U \cap Y$ contains infinitely many points of $Y$
However I know that the general topological definition of a limit point in a topological space is the following
Let $X$ be a topological space and let $Y \subseteq X$. A point $x \in X$ is a limit point of $Y$ if every neighborhood of $x$ contains at least one point of $Y$ different from $x$ itself.
I'm really curious as to why my lecturer defined a limit point in the way he did. Nothing in the definition of a metric space states the need for infinitely many points, however if we use the definition of a limit point as given by my lecturer only metric spaces that contain infinitely many points can have subsets which have limit points.
Wikipedia says that the definitions are equivalent in a $T_1$ space. The natural question to ask then would be are all metric spaces $T_1$ spaces?
Furthermore any finite metric space based on the definition my lecturer is using, would not have any subsets which contain limit points. Am I correct in saying this?