Problem 1
A continuous image of a compact space is compact; in Hausdorff spaces compact subsets are closed.
Faculty solution guide
Continuous Images and Extreme-Value Consequences
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 Continuous Images and Extreme-Value Consequences.
Problem 4
The argument must address using compactness to prove closedness, boundedness, and extrema 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.