To classify different types of topological spaces, mathematicians use separation axioms (often denoted as ). Long systematically walks through these layers: T1cap T sub 1
limits. Long generalizes this by defining a function as continuous if the inverse image of every open set is open. The chapter culminates in the definition of a —a bijective, bi-continuous map that proves two topological spaces are structurally identical. 4. Separation Axioms
You can find a PDF copy of by Paul E. Long (1971) at the Internet Archive . This site allows you to borrow the digital book or access restricted files with an account. The "Story" of the Book
For broader study, these supplementary resources provide concise overviews of general topology topics: an introduction to general topology paul e long pdf link
Spaces where you can measure the distance between points.
Paul E. Long’s An Introduction to General Topology is a well-regarded, upper-undergraduate level textbook. It strikes a balance between rigorous proof-based mathematics and clear exposition. Compared to heavier classics like Munkres or Kelley, Long’s book is often praised for being more accessible to a first-time learner while still covering essential topics thoroughly.
The text introduces abstract topological concepts by referencing familiar metric space ideas, creating a smooth transition from concrete to abstract mathematics. The chapter culminates in the definition of a
For many students and instructors, locating a digital copy of this textbook is a priority for coursework, research, or self-study.
Do you need assistance finding a or concept explanation? Do you need help solving a particular topological proof ?
If you are on a budget, there are exceptional open-access textbooks that can serve the same purpose. These resources can also be used alongside Long's book to provide different perspectives. Long (1971) at the Internet Archive
Because this is a classic text from 1971, it is widely available through digital libraries and academic archives:
The final major modules of the text focus on global topological properties:
The textbook standardizes the foundational topics required for any advanced study in mathematics, breaking them down into digestible, proof-heavy chapters: