Problem 1
Declare a set open when each point contains a metric ball lying inside the set; the resulting family satisfies the topology axioms.
Faculty solution guide
Equivalent Metrics and Topological Invariance
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 Equivalent Metrics and Topological Invariance.
Problem 4
The argument must address showing distinct metrics can determine the same open sets with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in analysis, optimization, computational geometry, function spaces, and scientific models, 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.