The study of algebraic Invariants of Topological spaces. This is because algebraic structures are easier to compare, so it helps for comparing topological spaces.
The notion of equivalence we focus to find invariants is based on notions of equivalence for loops in the topological space. In particular two: homotopic equivalence, and homologic equivalence.
Loops are maps (Continuous paths). Loops are equivalent when there's a continuous deformation between them.
One problem is that they are hard to compute. All the homotopy groups are not known even for the sphere!
Simplicial approximation theorem
Loops are subspaces (the images of Continuous paths for e.g.) Loops are considered equivalent if they bound a surface between them, basically.
Homological algebra, Homology long exact sequence
Some common operations on Topological spaces used in topology and particularly, algebraic topology. See vid: constructions
We can define products, taking the Cartesian product of the cells. There are some subtleties with the weak topology not coinciding with the product topology, but these only arise for infinite CW complexes..