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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 “topology”

“Connected ⇒ path-connected”

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

“Every subspace of a separable metric space is separable — fails for general topological spaces”
“Path components are closed”
“cl(A ∩ B) = cl(A) ∩ cl(B)”

False — A=Q,B=RA = \mathbb{Q}, B = \mathbb{R}∖Q\mathbb{Q} in R\mathbb{R}

“A product of normal spaces is normal”
“A continuous bijection is a homeomorphism”

False — id : (R(\mathbb{R}, discrete)→(R) \to (\mathbb{R}, usual), or t ↦ (cos t, sin t) from [0,2π)[0, 2\pi) to S1S^{1}

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

False — The unit ball of ℓ2\ell^{2}

“Connected components are open”

False — Q\mathbb{Q} with the usual topology

“A totally disconnected space is discrete”
“A compact T1T_{1} space is Hausdorff”

False — An infinite set with the cofinite topology

“An arbitrary intersection of open sets is open”

False — ∩n≥1(−1/n_{n\ge1}(-1/n, 1/n) = {0}