A countable dense subset of $\mathbb{R}$ can be $a G_\delta$

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

• Completeness is a topological property
• d(Tx, Ty) < d(x, y) for all x ≠ y on a complete space ⇒ T has a fixed point

Watch it happen

Baire category, and the argument it powers