Things often have a shape (continued from Geometry). Topology looks at the "raw" shape, the shape that is invariant under continuous transformations. The standard notion of "shape" is really Geometry (which includes topology).
Topology looks at neighbourhoods, what things are close to each other
It's like a generalization of Analysis
definition of a topology on a Set (see Topological space)
analytic topology notes https://courses.maths.ox.ac.uk/node/view_material/39341
Topology & Geometry - LECTURE 01 Part 01/02 - by Dr Tadashi Tokieda
A Hole in a Hole in a Hole - Numberphile