Problem 1
Find two disjoint nonempty open subsets whose union is the whole space.
Faculty solution guide
Connectedness and Separation
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Connectedness and Separation.
Problem 4
The argument must address proving a space cannot be partitioned into disjoint nonempty open sets with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in existence theorems, optimization, surface construction, periodic models, and boundary identification, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.