Counterexample bank for CSIR NET Maths Part C

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

“Connected ⇒ path-connected” — false

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

Connected as the closure of a connected set; no path reaches the segment.

connectednesstopology

#2 · Complex Analysis, Algebra & Topology › Topology

“Every subspace of a separable metric space is separable — fails for general topological spaces” — false

Counterexample locked — unlock with Notes + PYQ

topologyseparability

#3 · Analysis & Linear Algebra › Metric Spaces

“Path components are closed” — false

Counterexample locked — unlock with Notes + PYQ

connectednesstopology

#4 · Complex Analysis, Algebra & Topology › Topology

“cl(A ∩ B) = cl(A) ∩ cl(B)” — false

Counterexample: A = ℚ, B = ℝ∖ℚ in ℝ

cl(A ∩ B) = cl(∅) = ∅ but cl A ∩ cl $B = R$ .

topology

#5 · Complex Analysis, Algebra & Topology › Topology

“A product of normal spaces is normal” — false

Counterexample locked — unlock with Notes + PYQ

topologyseparation

#6 · Complex Analysis, Algebra & Topology › Topology

“A continuous bijection is a homeomorphism” — false

Counterexample: id : (ℝ, discrete) → (ℝ, usual), or t ↦ (cos t, sin t) from [0, 2π) to S¹

Needs compact domain and Hausdorff codomain.

topology

#7 · Complex Analysis, Algebra & Topology › Topology

“Countably compact implies compact” — false

Counterexample locked — unlock with Notes + PYQ

topologycompactness

#8 · Complex Analysis, Algebra & Topology › Topology

“The closed unit ball of a normed space is compact” — false

Counterexample: The unit ball of ℓ²

Riesz: compactness of the ball holds exactly in finite dimensions.

topologycompactness

#9 · Complex Analysis, Algebra & Topology › Topology

“Connected components are open” — false

Counterexample: ℚ with the usual topology

Components are singletons, which are not open. They are open exactly when the space is locally connected.

topologyconnectedness

#10 · Complex Analysis, Algebra & Topology › Topology

“A totally disconnected space is discrete” — false

Counterexample locked — unlock with Notes + PYQ

topology

#11 · Complex Analysis, Algebra & Topology › Topology

“A compact T₁ space is Hausdorff” — false

Counterexample: An infinite set with the cofinite topology

Compact and $T_{1}$ but any two non-empty open sets intersect.

topology

#12 · Analysis & Linear Algebra › Metric Spaces

“An arbitrary intersection of open sets is open” — false

Counterexample: ∩_{n≥1}(−1/n, 1/n) = {0}

Only finite intersections preserve openness.

topologyreal analysis