Counterexample bank

False — Closed unit ball in $\ell^{2} ($or in C[0,1] with sup norm)

False — $\mathbb{R}$ with the discrete metric

False — (0,1) and $\mathbb{R}$

False — Topologist's sine curve {(x, sin 1/x) : 0 < x ≤ 1} ∪ {0}×[−1,1]

False — f(x) = 1/x on (0,1)

False — $A_{4}$ has no subgroup of order 6

False — $\mathbb{Z}[\surd-5]: 6 = 2\cdot3 = (1+\surd-5)(1-\surd-5)$

False — y′ = $y^{1/3}$, y(0) = 0

False — y′ = $y^{1/3}$, y(0) = 0

False — $f(x) = x^{2}, g(x) = x$|x| on $\mathbb{R}$

False — $y'' + \pi^{2}y = f$ on [0,1] with y(0) = y(1) = 0

False — ẋ $= -y - x^{3}$, ẏ $= x - y^{3}$ at the origin

False — Two fair coin tosses: A = first is heads, B = second is heads, C = the two agree

False — The standard Cauchy distribution

False — $\max(X_{1}, X_{2})$ with $X_{i}$ ~ Exp(1)