Topological reasoning—the ability to identify structural invariants under continuous deformations—is a cornerstone of human visual cognition, yet it remains under-explored in modern AI systems. Existing benchmarks provide limited coverage of mathematical topology as a discipline and rarely probe the depth of topological reasoning. To bridge this gap, we introduce TopoEval, an 800-item benchmark suite for topological reasoning. TopoEval centers on a 400-item answer-based Core track and includes a 400-item Proof track for reference-grounded informal proof assessment. The suite spans four major topology branches, twelve fine-grained subfields, and three difficulty levels. On Core, we investigate model scaling, reasoning depth, prompt strategies, image dependency, and academic tiers. The strongest Core configuration reaches 73.0% overall and 54.1% on all 85 K12 items. For context, topology-trained graduate-student solvers reach 96.3% under the audited protocol in Appendix B.2. On Proof, pass-plus-borderline rates are low across the selected conditions, indicating difficulty in producing reference-aligned natural-language topology proofs. TopoEval profiles these capabilities across academic level, image dependency, and proof quality.