Is it possible to get a neighborhood with only finitely many points in it, in an infinite set? If we have an infinite set, is it possible to find a neighborhood around a certain point in the set that has only finitely many points in it?

Let $(x_k)$ be a convergent sequence of real numbers with limit $x$ not belonging to the sequence itself. Take the topology induced by the usual topology on the reals on $(x_k)$. Then every $x_k$ has an open neighborhood consisting of $x_k$ only. Every $x_k$ sufficently near to $x$ also has an open neighborhood consisting of infinitely many points, but still different from the whole space.
