The bookUnit 1 · Metric Spaces16 / 83

Completeness and Baire category

Why this is asked: Completeness: closed subsets of complete spaces, ℓᵖ, C[0,1] with sup norm are complete; ℚ, (0,1), C[0,1] with L¹ norm are not. Baire: ℝ is not a countable union of nowhere dense sets; a complete metric space without isolated points is uncountable.

Completeness, Banach fixed point and Baire category

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See it move

Baire category, and the argument it powersinteractive

Why ℝ is not a countable union of nowhere dense sets, and how that one fact proves several others.

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The trap here

“Completeness is a topological property” — false

(0,1) and $\mathbb{R}$

Homeomorphic, but $\mathbb{R}$ is complete and (0,1) is not (1/n is Cauchy).