Open Sets

About

This is a personal notebook. The entries are short — often a single proof, a counterexample, or a worked exercise — written for my own future reference. The content drifts among:

• point-set topology (continuity, compactness, connectedness, separation axioms)
• metric spaces and the topology that survives the metric being thrown away
• elementary algebraic topology (fundamental groups, covering spaces, simple homology)

Nothing here is original. Most posts are paraphrases of Munkres, Bredon, or Hatcher, occasionally with a small numerical sanity-check or a hand-drawn picture redrawn in tikz.

Recent notes

• 2026-04 Why the long line is not metrizable — a short, careful argument — separability fails because uncountably many disjoint open intervals can be embedded.
• 2026-03 The Tychonoff product theorem, two ways — finite-product proof from compactness, then the Zorn / ultrafilter argument for the general case.
• 2026-02 Quotient spaces of the disk and what gluing actually means — a careful walk through how the torus, Klein bottle, and projective plane arise from squares.
• 2026-01 Connected components vs. path components — when they disagree — the topologist's sine curve, with a clean proof that it is connected but not path-connected.
• 2025-12 Covering spaces of the figure-eight — a tree of subgroups — sketching the lattice of subgroups of the free group on two generators via covering spaces.
• 2025-11 Why "every metric space is normal" is harder than it looks — the standard proof, the usual mistake, and why the standard proof actually works.

Reading list

Working through Munkres's Topology as the spine, with Bredon's Topology and Geometry for a more category-flavoured retake on the algebraic chapters. Hatcher's Algebraic Topology is the long-term goal — I dip in occasionally when a Munkres exercise raises a question that won't be answered until much later.

Contact

Corrections welcome at opensets at fundamentalchaos.xyz. I read mail irregularly but I do reply eventually.