Problem 1
Check that ∅ and X lie in τ, arbitrary unions of members of τ lie in τ, and finite intersections of members of τ lie in τ.
Faculty solution guide
Topological Spaces and the Open-Set Axioms
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 Topological Spaces and the Open-Set Axioms.
Problem 4
The argument must address verifying arbitrary unions, finite intersections, and the empty and whole sets with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in configuration spaces, data neighborhoods, continuity, coordinate systems, shape comparison, and mathematical modelling, 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.