Topology is where mathematics gets both more abstract and, paradoxically, more visual — it studies the properties of space that survive stretching and bending. The difficulty is that the subject splits into two very different-feeling halves: the careful, set-theoretic point-set topology, and the sweeping, algebraic machinery of algebraic topology. Trying to appreciate the second before you are solid in the first is the classic way to get lost.
The order that works builds the point-set foundations rigorously, transitions into algebraic topology through a well-paved on-ramp, and then reaches the advanced structures that modern topology and geometry rest on. Each layer supplies the language the next one assumes.
Point-set foundations
Start with Topology by Munkres, the near-universal first course, whose clear exposition of open sets, continuity, compactness, and connectedness makes it the standard for a reason. General topology by Kelley is the more advanced classic reference for the same material when you want depth on the harder theorems. Topology by Jänich is a lovely shorter alternative, prized for its intuition and readability, that many use to build a feel for the subject alongside the rigor. This foundation is non-negotiable for everything after.
Into algebraic topology
The transition is the crux. A basic course in algebraic topology by Massey is a gentle, well-structured introduction to the fundamental group and homology that eases you across the divide. Algebraic Topology by Hatcher is the modern standard-bearer — comprehensive, geometric, and freely available — and it becomes the reference you return to for years. Algebraic topology and transformation groups extends the ideas with group actions for readers ready to go further.
Advanced structures
The final arc is graduate-level machinery. Lectures on characteristic classes by Milnor is the classic on characteristic classes, a cornerstone of modern topology and geometry. Vector Bundles and K-Theory introduces the vector-bundle and K-theory perspective, and A User's Guide to Spectral Sequences teaches the computational engine that makes hard homology calculations tractable. For homotopy theory proper, Modern classical homotopy theory and Stable homotopy and generalised homology are the deep texts that define the frontier.
Read in this order and topology's two halves connect into one coherent subject. Follow the full path to go from open sets to stable homotopy theory.
