Deleting $n$ points from a connected space

Question

Let $X$ be a space such that for any subset $S \subset X$ with finite cardinality $n$, the subspace $X \setminus S$ has exactly $n+1$ connected components, each of which is homeomorphic to $X$. Is there such a space $X$ which is not homeomorphic to $\mathbb{R}$?

Answer 1

Here's a bit of a cheap example. Consider the topology on $\mathbb{R}$ described in this previous answer. Since the topology is finer than the usual topology, the removal of any point disconnects the space, and it is relatively easy to see that there are two connected components, each of which is homeomorphic to the original space.