Difference between revisions of "Topology Qualifying Syllabus"
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===Topological spaces and continuous functions:=== | ===Topological spaces and continuous functions:=== | ||
Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology. | Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology. | ||
| + | |||
| + | [[Quotients (Topology)]] | ||
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===Homeomorphism invariants:=== | ===Homeomorphism invariants:=== | ||
Separation properties (T0, T1, Hausdorff, regular, normal), | Separation properties (T0, T1, Hausdorff, regular, normal), | ||
countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications. | countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications. | ||
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[[Path Connected]] | [[Path Connected]] | ||
| + | [[ T2 (Hausdorff)]] | ||
| + | [[Compact]] | ||
===Continuous deformations:=== | ===Continuous deformations:=== | ||
Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type. | Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type. | ||
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==Algebraic topology:== | ==Algebraic topology:== | ||
===Fundamental groups:=== | ===Fundamental groups:=== | ||
Latest revision as of 22:02, 21 May 2023
Contents
Point-set Topology:
Topological spaces and continuous functions:
Topology, open and closed sets, basis, subbasis; continuous function, homeomorphism; closure, limit points; subspace topology, product topology, and quotient/identification topology.
Homeomorphism invariants:
Separation properties (T0, T1, Hausdorff, regular, normal), countability properties; connectedness, path connectedness, components; compactness, metrizability. Applications.
Path Connected T2 (Hausdorff) Compact
Continuous deformations:
Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.
Algebraic topology:
Fundamental groups:
Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.
Covering spaces:
Covering map, Lifting theorems; covering space group action; universal covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal covering; applications to group theory.
Homology:
Simplicial homology, singular homology, induced homomorphism, homotopy invariance; exact sequence, long exact homology sequence, Mayer-Vietoris Theorem. Applications.
