(a) Let $X$ be a connected topological space such that each point $x$ of $X$ has a neighbourhood homeomorphic to $\mathbb{R}^{n}$. Prove that $X$ is path-connected.

(b) Let $\tau$ denote the topology on $\mathbb{N}=\{1,2, \ldots\}$, such that the open sets are $\mathbb{N}$, the empty set, and all the sets $\{1,2, \ldots, n\}$, for $n \in \mathbb{N}$. Prove that any continuous map from the topological space $(\mathbb{N}, \tau)$ to the Euclidean $\mathbb{R}$ is constant.