Topology is often called "rubber-sheet geometry". It asks what stays the same about a shape when you stretch, bend and twist it — but never tear or glue. To a topologist a coffee mug really is a doughnut: each has exactly one hole, and you can mould one into the other without cutting. Distance, angle and straightness all melt away; what survives is a shape's deepest, most stubborn structure — its connectedness, its holes, its nearness.
To make "nearness without distance" precise, topology builds everything from a single, spare idea: the open set. From that one notion flow continuity, limits, compactness and connectedness — the load-bearing beams of modern analysis, geometry and even parts of physics and data science.
One thread runs through everything here. Decide which subsets of a space count as open, and you have declared what "close" means without ever measuring a distance. A map is continuous when it never rips that structure apart, and two spaces are the same (homeomorphic) when a continuous, invertible deformation carries one to the other. Every theorem in the course is really about what such deformations can and cannot do.
This course moves in two stages, from the definitions to the great theorems.
It all begins by answering one question with surprising care: what, exactly, is a "space"? The answer — a collection of open sets — is the seed from which the whole subject grows.