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Course Syllabus
Talk 1	Introduction: what does topology studies?
	Part	one: topological spaces
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
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
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
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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