TITLE: Why $\mathbb Q$ is not locally compact, connected, or path connected?
QUESTION [22 upvotes]: Why is the set of rationals $\mathbb{Q}$, as a subspace of $\mathbb{R}$ with the usual (Euclidean) topology, is not locally compact nor locally connected not locally path connected ??

REPLY [7 votes]: Another argument for the local compactness : it's not locally compact, because it's not a Baire space, and all locally compact Hausdorff spaces are Baire spaces.
The components of $\mathbb{Q}$ are singletons (because between every two rationals lies an irrational that separates them, essentially), which kills connectedness, and every open subset of $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}$, and so this also holds for all open subsets. So the space is not locally connected (which is equivalent to every component of an open subset is open), because $\mathbb{Q}$ has no isolated points.