Consider a finite $T_1$ topological space $(X,T)= \{x_1, x_2,...x_n\}$ . According to the definition of a $T_1$ space, every singleton is a closed set. I am wondering if you can have limit points in this type of space.
My attempt: Suppose $x_1$ is a limit point of $(X,T)$, then since B:=$\{x_2,...x_n \}$ is a closed set (a finite union of closed sets), the complement of $B$ is an open set containing $x_1$ that contains no other elements of the topology, therefore $x_1$ is not a limit point, a contradiction. Am I on the right track?
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As Theo Bendit commented, you are right. You have shown that all one-point subsets are open, i.e. that $X$ is discrete. In a discrete $X$ no point is a limit point.