Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Analysis & Linear Algebra › Metric Spaces

“Closed and bounded ⇒ compact” — false

Counterexample: Closed unit ball in ℓ² (or in C[0,1] with sup norm)

$e_{1}, e_{2}, \dots$ has no convergent subsequence since ‖ $e_{n} - e_{m}$ ‖ $= 2$ . Heine–Borel is $R^{n} -$ only.

compactnessmetric spaces

#2 · Analysis & Linear Algebra › Metric Spaces

“Bounded ⇒ totally bounded” — false

Counterexample: ℝ with the discrete metric

Everything is within distance 1, but no finite set of balls of radius ½ covers it.

compactnessmetric spaces

#3 · Analysis & Linear Algebra › Metric Spaces

“Completeness is a topological property” — false

Counterexample: (0,1) and ℝ

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

completenessmetric spaces