Connected as the closure of a connected set; no path reaches the segment.
Counterexample bank
Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.
AllODEPDEabeliananalyticityanovabairecalculus of variationscanonical formscauchy-riemanncompactnesscompletenesscomplexconformalconnectednesscontinuitycontour integrationconvergencedifferentiationdistributionsfieldsfixed pointgaloisgreengroupsharmonicinjectivityintegral equationsintegrationlaurentlimsuplinear algebraliouvillelpmarkovmeasuremechanicsmetric spacesmultivariablemultivariatenumericalp-groupspermutationspolynomialspower seriesprobabilityquadratic formsreal analysisregressionresiduesringssamplingseparabilityseparationsequencesseriessingularitiesstabilitystatisticssylowtestingtopologyuniform convergencewronskianzeros
#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]
connectednesstopology
#2 · Analysis & Linear Algebra › Metric Spaces
“Closure of a path-connected set is path-connected” — false
Counterexample: Graph of sin(1/x) on (0,1]
Its closure is the topologist's sine curve.
connectedness
#3 · Analysis & Linear Algebra › Metric Spaces
“Path components are closed” — false
Counterexample locked — unlock with Notes + PYQ
connectednesstopology
#4 · 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