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Counterexample bank

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

All ODE PDE abelian analyticity anova baire calculus of variations canonical forms cauchy-riemann compactness completeness complex conformal connectedness continuity contour integration convergence differentiation distributions fields fixed point galois green groups harmonic injectivity integral equations integration laurent limsup linear algebra liouville lp markov measure mechanics metric spaces multivariable multivariate numerical p-groups permutations polynomials power series probability quadratic forms real analysis regression residues rings sampling separability separation sequences series singularities stability statistics sylow testing topology uniform convergence wronskian zeros

#1 · 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

#2 · Analysis & Linear Algebra › Metric Spaces

“d(Tx, Ty) < d(x, y) for all x ≠ y on a complete space ⇒ T has a fixed point” — false

Counterexample: T(x) = x + 1/x on [1, ∞)

Distances strictly decrease but no contraction constant k < 1 exists; no fixed point.

completenessfixed point

#3 · Analysis & Linear Algebra › Metric Spaces

“A countable dense subset of ℝ can be a G_δ” — false

Counterexample locked — unlock with Notes + PYQ

completenessbaire