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Course Syllabus |
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Talk 1 |
Introduction: what does topology studies? |
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Part |
one: topological spaces |
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| Talk 2 | - | Topological structure defined by open and closed sets |
| Talk 3 | - | Other basic concepts: neighborhood, interior, closure, etc. |
| Talk 4 | - | Example: metric space |
| Talk 5 | - | Continuous mapping and homeomorphism |
| Talk 6 | - | Product space |
Part |
two: Point set topology |
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| Talk 7 | - | Separation axioms and countability axioms |
| Talk 8 | - | Urysohn lemma, Tietze extension theorem, and metrization theorem |
| Talk 9 | - | Compactness and sequentially compactness |
| Talk 10 | - | Connectedness |
| Talk 11 | - | Path and pathwise connectedness |
| Talk 12 | - | Example: finding non-homeomorphic topological structures on R |
Part |
three: simple topological manifolds |
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| Talk 13 | - | Cut-and-paste operation and quotient space |
| Talk 14 | - | Mobius strip and projective plane |
| Talk 15 | - | Topological manifold, closed surface |
| Talk 16 | - | Classifying and identifying closed surfaces |
| Talk 17 | - | What can we say about 1-manifolds: knots and links |
| Talk 18 | - | Example: 3-dimensional sphere |
Part |
four: fundamental groups |
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| Talk 19 | - | Homotopy and mapping class |
| Talk 20 | - | Fundamental groups |
| Talk 21 | - | Example: fundamental groups of circle and sphere |
| Talk 22 | - | Homotopic equivalence, deformation retract |
| Talk 23 | - | An outline of finitely presented groups, van Kampen theorem |
| Talk 25 | - | Classical applications of fundamental groups: Brouwer fixed point theorem in dimension 2, fundamental theorem of algebra, etc. |
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