Topology I 1000-113bTP1a

1. Metric spaces. Topology of metric spaces. Topological spaces. Base of a topology. Interior and closure of a set, subspaces. Hausdorff spaces. Continuous mappings, characterizations of continuity. Homeomorphisms. Tietze Theorem on extensions of mappings (for metrizable spaces). Cartesian products of topological spaces. Separable spaces. (3 lectures)
2. Compact spaces. Conditions equivalent to compactness in metrizable spaces. Compact subsets of Euclidean spaces. Continuous mappings on compact spaces. Weierstrass Theorem. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Uniform continuity. The Cantor set. Tychonoff
Theorem on compactness of Cartesian products of compact spaces (proof for finite products). (3 lectures)
3. Complete metric spaces. If a metric space $Y$ is complete then the space of bounded continuous functions $C_{b}(X,Y)$ equipped with the sup metric is complete. Banach Fixed Point Theorem. Baire Theorem. Metric space is compact iff it is complete and totally bounded. Ascoli-Arzeli Theorem. (2 lectures)
4. Connected spaces. Path connectedness. Components and path components. (1 lecture)
5. Homotopic mappings. Contractible spaces. Homotopic loops. Simply connected spaces. Proof of the noncontractibility of the circle. Corollaries: there is no retraction from a disc onto its boundary circle, Brouwer Fixed Point Theorem for dimension 2. Proof of the Fundamental Theorem of Algebra. (3 lectures)
6. Quotient spaces. Attaching a space $Y$ to $X$ along $A subset Y$ via $f : A to X$. Two-dimensional manifolds. Examples of surfaces obtained by identification of edges of regular polygons. (2 lectures).