# Definition of a limit point in a metric space

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?

• Every matrix space is a $T_1$ space since for $x,y\in X$ with $d=d(x,y)$ the neighborhoods $B(x,d/2)$ and $B(y,d/2)$ separate $x$ and $y$. Commented Mar 10, 2018 at 21:18
• For your last question in your post, you are correct. This can be seen using the definition the other definition too. Let x be a point and consider the open ball with center x and radius the minimum of all distances to other points. Then, this ball only contains x. Hence, x is not a limit point. Since x was arbitrary, there are no limit points.
– user370967
Commented Mar 10, 2018 at 22:05
• Metric spaces are $T_n$ spaces for $n\in \{ 0,1,2, 2\frac {1}{2}, 3, 3\frac {1}{2},4,5,6 \}.$ Commented Mar 13, 2018 at 1:10

As said in comments, both definitions are equivalent in the context of metric spaces. If one point can be found in every neighborhood, then, after finding such a point $x_1$, we can make the neighborhood smaller so that it does not contain $x_1$ anymore; but there still has to be a point in there, say $x_2$,... the process repeats.
The situation is different in weird topological spaces that are not $T_1$ spaces. For example, if X is a space with trivial topology, then for every nonempty subset $Y\subset X$ (even a finite one), every point $x\in X$ is a limit point. Indeed, there is only one neighborhood of $x$, namely the space $X$ itself; and this space contains a point of $Y$. This example shows that in non $T_1$-spaces two definitions are no longer equivalent. The second one is to be used in this case.