Problem 1
U⊆X/~ is open exactly when its preimage under the quotient map is open in X.
Faculty solution guide
Quotient Spaces, Identifications and Gluing
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 Quotient Spaces, Identifications and Gluing.
Problem 4
The argument must address building circles, cylinders, tori, and projective spaces from equivalence relations 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.