Università Roma Tre

Part 1. Topological spaces

Topology; euclidean, trivial, discrete, cofinite, cocountable topology .
Bases, local bases.
Sottospazi di topological spaces: closure, interior, boundary.
Closure, limit points
Continuous, open, closed maps.
Omeomorphism
Product of topological spaces.
T1 and Hausdorff.
Sequences and their limits.
N1 and N2 spaces
separable spaces .
Quotient topology .

Part 2. Connectedness compactness and metric spaces

Connectedness, connected components
Arcs and arc connectedness
Compactness.
Tychonoff' theorem (proof in the finite case only ).
Distance and metric spaces
Separation properties: T1, T2, T3, T4.
Metrizable spaces,
Cauchy sequences in metric spaces.
Complete and compact metric spaces, Lebesgue number.

Part 3. Homotopy and fundamental group

Homotopy di continuous maps .
Contractible spaces.
Homotopy oftopological spaces.
Arcs and loops: products of arcs and equivalence of arcs.
Fundamental group.

Covering spaces topological spaces.
Fundamental group of the circle.
Lifting of continuous maps to covering spaces: existence and uniqueness.
Classification of covering spaces via the fundamental group.
Homopic invariance of the fundamental group.
Seifert-Van Kampen Theorem.
Fundamental group of spheres.

James R. Munkres Topology Prentice Hall.

Lecture notes corso by the teacher available on line.