MATH 562- TOPOLOGY II-SPRING 2026- A. FREIRE

Syllabus, references and topics

Tu 1/20  Topologies on spaces of maps (ref: [Munkres, sect. 46]

Compactness, countability, function spaces
(review Ex. 1,2, 5, 6, 7, 8, 9 of handout)

Th 1/22  Function spaces: examples
Ex. 3, 4 and 10, 11 of the above handout

Tu 1/27 [snow day] u.o.c topology: metrizability, separability
(Ex 12,13,14 of handout)

Th 1/29: Arzela-Ascoli and applications (start)
Arzela-Ascoli notes

Problem Set 1

Tu Feb 3: Arzela-Ascoli: proof, applications (end)

Th Feb 5: Stone-Weierstrass Theorem (for algebras, lattices)

Stone-Weierstrass notes

Tu Feb 10: Weierstrass thm/ Baire spaces/ meager and generic sets, Baire's theorem

Th Feb 12: discontinuity sets of pointwise limits are meager/nowhere-diff'ble fns generic (outline)/
diff'ble manifolds (start)

Problem Set 2

Tu Feb 17 Manifolds arising as quotients: Hausdorff, second countability. Examples:
Sn (stereographic coords), RPn (n+1 charts). Tangent space at a point.

Th Feb 19 Differential of a smooth map. Tangent bundle, vector bundles. Grassmannians
as quotient manifolds (of Stiefel manifolds)

Tu Feb 24 Smooth atlas for Grassmannians/Submanifolds/Local form of immersions.

Problems [GP-p. 18:]  6(a)(b), 8 (show that the map given is a proper map R--> R^2, and an injective immersion)

Notes on manifolds
(Current to 2/26; includes exercises.)

Th Feb 26: Injective immersions vs. embeddings (examples); proper maps/ local form of submersions/
preimages of regular values are submanifolds

Problems [GP--p. 25:] 1, 2, 6, 8, 11, 12

Tu Mar 3 Orthogonal groups (example)/Transversality/connection with regular values

Th Mar 5 Partitions of unity and applications

Tu Mar 10, Th Mar 12: SPRING BREAK

next: openness of immersions, submersions, etc. /Sard's theorem