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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?

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    $\begingroup$ 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 $y$. $\endgroup$ – Levent Mar 10 '18 at 21:18
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    $\begingroup$ 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. $\endgroup$ – user370967 Mar 10 '18 at 22:05
  • $\begingroup$ Metric spaces are $T_n$ spaces for $n\in \{ 0,1,2, 2\frac {1}{2}, 3, 3\frac {1}{2},4,5,6 \}.$ $\endgroup$ – DanielWainfleet Mar 13 '18 at 1:10
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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.

I prefer the second definition myself, but the first definition can be useful too, as it makes it immediately clear that finite sets do not have limit points.

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.

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