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So my book says that if $A\subset X$ is finite then $A'=\emptyset$ does that go backwards aswell? meaning that if $A'=\emptyset$ does that mean that the set is finite?

I have an exercise to do that wants me to prove that if $A'=\emptyset$ then prove that $A$ is closed and i believe i could do something if that is a thing.

NickSorgas
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    I don't know much about the topic here, but wouldn't simply $S = \mathbb{Z}$ be a counterexample? – Stephen Donovan Apr 30 '21 at 23:13
  • If every point in the complement of $A$ is not a limit point of $A$, then what can you find for each of those points? – Joe Apr 30 '21 at 23:22
  • @Joe means the point is isolated therefore a sigleton? – NickSorgas Apr 30 '21 at 23:26
  • No, I’m not talking about isolated points of $A$. I’m talking about points in the complement of $A$. If each of the points in the complement of $A$ are not limit points of $A$, what can we find for each of those points? – Joe Apr 30 '21 at 23:35
  • @Joe I cant quite get what you are trying to tell me I probably have to study more thanks for the effort though. <3 – NickSorgas Apr 30 '21 at 23:39
  • I’m happy to help. If you look at the definition of limit point, you’ll see that to be a limit point “something” must be true about all... And since each point in the complement of $A$ is not a limit point, “that thing” is not true about all... – Joe Apr 30 '21 at 23:54

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A useful theorem about limit points is that: If $X$ is any topological space and $A \subset X$ is a closed subset, then $A = A'$, meaning that a closed set contains all its limit points. So, if $A' = \emptyset $, then $A$ contains all its limit points and must be closed.

colbyjack101
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