1.2 Introduction to Algebraic Topology. CW Complexes.

Among all topological spaces, CW complexes are the first approximation to the best possible spaces to study, using homotopy invariant algebraic techniques (the best ones are too difficult to define for the level of this course). Such complexes comprise a variety of spaces large enough to start to play and have some fun with homotopies on them. They are obtained by recursively attaching discs on lower dimensional ones via so called "attaching maps".
In particular, CW complexes are locally contractible and satisfy some cardinality bounds.