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?
 A: Take $\mathbb N$ for example. It is possible to find neighborhood around any point that has exactly $k$ elements in it, for every $k \in \mathbb N$
A: 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.
A: The concept of a neighborhood requires that we have a topology on our set. So if we just have an infinite set, then no we can't pick a neighborhood (finite or otherwise) because we don't know what those are.
If you want to know if we have a topological space of infinitely many points is it possible to find a finite neighborhood, then the answer is not always. For example: Let $X$ be an infinite set, then under the discrete topology, of course there is a finite neighborhood of a point $x$, the singleton $\{x\}$! On the other hand, when $X$ has the trivial topology, the only neighborhood of $x$ is $X$.
So no, it is not always possible, given an arbitrary topology on an infinite set.
