Here you will find the topics covered, lecture by lecture. A rough outline is as follows:

•   Topologies and continuous functions (axioms and basic properties).

•   Product topology, induced topology, quotient topology.

•   Separation properties, connectedness, compactness and compactification.

•   Other arguments, depending on students' interests.

Lecture 1: definition of topology and first examples on R, comparing two topologies, base of a topology; closed sets, interior and closure of a subset, relative topology on a subset. 

Lecture 2: continuous maps, homeomorphisms, open and closed maps. Product spaces: product topology and box topology. Quotient spaces and quotient maps; contraction of a subspace to a point, quotient by an equivalence relation, projective spaces.

Reference books:

• Topology, James R. Munkres: this is the main reference, it's freely available on the Internet Archive.

• General Topology, S. Willard: secondary reference, also available on the Archive, but more inconvenient to read.

Here you will find some notes and exercises.