Università Roma Tre

Topological spaces. Examples and first properties. Metric spaces and continuous applications. Examples. The topological space associated with a metric space.
Metrizable and non metrizable spaces, topologically equivalent metric spaces. Closed subsets of a topological space and closing operators, neighborhood and border of a subset of a topological space.
Properties and examples of closures, interiors and borders of sub-spaces of a topological space. Limit points and derived set of a subset of a topological space. Basis for a topology. Characterization of a base.
Sub-bases for a topology. Separable topological spaces. Basic bases of surroundings for a point and axioms of numerability. Limit of a sequence. Continuity of applications between topological spaces.
Examples of continuous applications. Open and closed applications. Homeomormisms between topological spaces. Examples of homeomorphic and non-homeomorphic spaces. The product topology.
Examples and properties of product topologies. The topology produced on infinite collections of topological spaces. The quotient topology. Examples of identifications.
The quotient topology: examples and characterization through open saturated sets. Hausdorff or T2 spaces: definition, examples and first properties.
Properties of Hausdorff spaces. T0 and T1 spaces: examples and properties. Regular and normal spaces.
Normal spaces. The lemma of Uryshon and the theorem of the extension of Tietze.
Lindelof spaces: definitions, properties and examples. Compact spaces: definition, examples and first properties. Compacts in Hausdorff spaces.
Continuous and compact applications. The Heine-Borel theorem and compact subspaces of R.
Tychonoff's theorem. Generalizations of compactness: locally compact, paracompact, numerically compact and compact spaces for sequences.
Equivalence between compact and complete metric spaces and totally limited. Compactification of Alexandroff and Stone-Cech.
Stone-Cech compactification: existence and properties. Connected spaces: definition and first properties.
Properties of connected spaces and applications, connected components and connection by arcs.
Connected components for arcs and properties. Homotopy of maps.
Homotopic equivalence, examples. Fundamental group.
Functorial properties of the fundamental group. Hints on the theory of categories.
Properties of the fundamental group: invariance for arc connected components and properties of the fundamental group of contractible spaces. The fundamental group of S^1 is isomorphic to Z.
Coverings. Theorem of the lifting of the homotopies and some consequences. Morphism induced by a covering in the fundamental groups.
Coverings and fundamental group. Monodromy. Correspondence between subgroups of the fundamental group and isomorphism classes of coverings.

E. Sernesi: Geometria 2, Bollati Boringhieri, 2001.
S. Willard: General Topology, Dover Publications, 2004.
J. L. Kelley: General Topology, Graduate Text in Mathematics, Springer 1975.
J. R. Munkres: Topology: a first course, Prentice Hall, 1974.
A. Hatcher: Algebraic Topology, Cambridge University Press, 2002.