This course is concerned with the study of topological spaces and their structure preserving maps (continuous maps). Topological methods and ideas are universal in modern mathematics. A topological space is a type of geometric object that does not necessarily have a distance function associated with it but, via the notion of open sets (introduced by axiomatic properties), it encodes the rigorous approach to the idea of "continuity" and "nearness". Many different examples of topological spaces will be examined, including a class of topological spaces called topological manifolds (spaces that locally look like a Euclidean space), and in particular 2-dimensional topological manifolds, called "surfaces". Examples of surfaces are the 2-sphere S2, R2, 2-torus ("the surface of a bagel") and the "Klein bottle". The course concludes with an introduction to combinatorial topology (which considers topological spaces by means of simplicial complexes and triangulations) and a discussion of the classification theorem for closed surfaces.
1. Topological spaces and continuous maps
From geometrical to topological properties: an intuitive idea of the topological equivalence. Recollection of continuous functions: continuity for a function on a real line; continuity at a point for a map of metric spaces; neighborhoods; open sets in metric spaces and their properties. Definition of a topological space. Alternative description for the continuity at all points. Properties of continuous maps. The notion of a category. Homeomorphism. Bases.
2. Topological constructions
Induced topology and subspaces. Definition of induced topology. Particular case: subspaces. Examples. Properties. Coinduced topology and identification spaces. Definition of coinduced topology. Equivalence relations, cosets (recollection). Identification spaces. Examples. Properties. Product spaces. Base of product topology. Product spaces: properties and examples.
3. Fundamental topological properties
Closed sets. Definition; properties (dual to the properties of open sets); a map is continuous if and only if the preimage of every closed set is closed; closure. Hausdorff property. Definition, examples, cases of subspaces and quotient spaces (different behaviour), product spaces. Compactness. Definition and general properties. Interaction with Hausdorff property: the "homeomorphism theorem". Examples of applications. Compactness in Rn: Heine-Borel Lemma (compactness of the cube) and Heine-Borel Theorem ("compact" in Rn is equivalent to "closed and bounded"). Connectedness and path-connectedness. Definition of connectedness and general properties. Interval is connected. Path-connectedness: general properties and relation with connectedness. Connected and locally path-connected space (e.g., an open subspace in Rn) is path-connected.
4. Manifolds and surfaces
Definition of a manifold. Examples. Transition functions. Differentiability. Orientation (for differentiable manifolds). Surfaces (2-dimensional manifolds). Examples. Classification of closed (= compact connected) surfaces.
5. Simplicial complexes and Euler characteristic
Simplices: examples in low dimension, definition. Finite simplicial complexes. Body. Triangulation of a topological space. Examples. Euler characteristic. Topological invariance. "Excision formula" (the Euler characteristic for the union of complexes). Examples of calculation. Application to classification of surfaces.
Theodore Voronov 7 January (20 January) 2006