definition of limit points of a set vs limit point of a sequence.

I found the definition below from Encyclopedia,

Defition: “In mathematics, a limit point (or cluster point or accumulation point) of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself”

I thought every neighborhood of “a limit point of a set X” must contain infinitely many terms of some sequence which converges to that limit point. As far as I understand “if there is a neighborhood of the limit point that only contains a finite number of points other than itself then it does not qualified to be a limit of any sequence in the set X” because for example on the real line, if there is a neighborhood of a limit point that contains only a finite number of points other than itself, it means that after a fixed term n of a sequence, there are only a finite number of terms.

Could someone please correct my misconception. Thank you so much.

• Using "limit" for all of these can be confusing. You will also see "accumulation point" and "cluster point" for various ones. Commented May 13, 2023 at 18:20

You are right if the space is Hausdorff (or even $$T_1$$). But otherwise it's perfectly possible to construct examples where the number of points is finite. Take for example $$\{1,2\}$$ with the trivial topology. Then $$2$$ is a limit point for $$\{1\}$$.
• Why should $2$ be an isolated point? Commented May 13, 2023 at 18:44
• @HuyenLinhHo, not at all, since $\{2\}$ is not open. Only $\{1,2\}$ and $\emptyset$ are open in the trivial topology on $\{1,2\}.$ In fact, it would not be possible for a point to be isolated and a limit point. In order for a point $x$ to be isolated in a topology on $X,$ $\{x\}$ must be open in said topology. Then $\{x\}$ is a neighborhood of $x$ that contains no other point from any set, so can't be a limit point of any subset of $X.$ Commented May 13, 2023 at 19:06