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CSIR NET Part C

Counterexample bank

Part C is won by knowing which tempting claims are false. 149 counterexamples; 86 free to read right now. Pick your subject below.

Knowing the claim is false is half of it. The mistakes that cost marks shows the wrong options candidates actually pick, and the reasoning slip behind each one.

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Tagged “compactness”

“Closed and bounded ⇒ compact”

False — Closed unit ball in ℓ2(\ell^{2} (or in C[0,1] with sup norm)

“Bounded ⇒ totally bounded”

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

“Countably compact implies compact”
“The closed unit ball of a normed space is compact”

False — The unit ball of ℓ2\ell^{2}

“A uniformly bounded sequence in C[0,1] has a uniformly convergent subsequence”

False — fn(x)=xnf_{n}(x) = x^{n}