Topology
Shape without distance: how continuity depends only on open sets, topological spaces defined by open sets with bases, subbases, closures, interiors and subspaces, and examples from metric, discrete, cofinite, Sierpiński and lower limit topologies, continuous maps, the pasting lemma and homeomorphisms such as stereographic projection, product and quotient topologies with their universal properties and the torus, Möbius strip, Klein bottle and projective plane glued from a square, the separation axioms with Urysohn's lemma, the Urysohn metrization theorem and the non-normal Sorgenfrey plane, connectedness of intervals, path-connectedness and the topologist's sine curve, compactness with the tube lemma, the Heine–Borel theorem and continuous bijections from compact to Hausdorff spaces, Tychonoff's theorem proved with maximal families and Zorn's lemma, local compactness and the one-point compactification, homotopy and the fundamental group, the fundamental group of the circle by path lifting with the Brouwer fixed point theorem, the fundamental theorem of algebra and the Borsuk–Ulam theorem in dimension 2, covering spaces and the Seifert–van Kampen theorem, and a capstone classifying closed surfaces by orientability and Euler characteristic, with runnable Python examples throughout.
MATH-447 Year 4 3 credits 12 lessons · 5 h 19 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- Explain how continuity in metric spaces depends only on open sets, distinguish topological from metric properties, and prove that spaces are not homeomorphic using invariants.
- Work with topological spaces through bases, subbases, closures, interiors and subspaces, using metric, discrete, cofinite, Sierpiński and lower limit topologies as examples.
- Prove that maps are continuous using preimages, closures, subbases and the pasting lemma, and construct or rule out homeomorphisms such as stereographic projection.
- Build product and quotient spaces, apply their universal properties, and describe the torus, Möbius strip, Klein bottle and projective plane as quotients of a square.
- State and apply the separation axioms, prove that metric spaces are normal and that limits are unique in Hausdorff spaces, and use Urysohn’s lemma and the Urysohn metrization theorem.
- Prove that intervals are connected, that connectedness is preserved by continuous images, closures and products, and distinguish connectedness from path-connectedness using the topologist’s sine curve.
- Prove the basic properties of compact spaces, including the tube lemma, the Heine–Borel theorem and the fact that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
- Prove Tychonoff’s theorem using maximal families with the finite intersection property, and construct one-point compactifications of locally compact Hausdorff spaces.
- Define homotopy and the fundamental group, prove its group properties and invariance under homeomorphism and homotopy equivalence, and compute it for convex sets, products and spheres.
- Compute the fundamental group of the circle by path lifting and apply it to prove the Brouwer fixed point theorem in the plane, the fundamental theorem of algebra and the Borsuk–Ulam theorem in dimension 2.
- Use covering spaces and the Seifert–van Kampen theorem to compute fundamental groups of the projective plane, wedges of circles, punctured planes and surfaces.
- Classify closed surfaces by orientability and Euler characteristic, compute these invariants from polygon words and triangulations, and outline the proof of the classification theorem.
Course content
12 lessons · 5 h 19 min total · final exam
- 1 Why topology: shape without distance 26 min · quiz
- 2 Topological spaces, bases and subspaces 27 min · quiz
- 3 Continuous maps and homeomorphisms 25 min · quiz
- 4 Product and quotient topologies 27 min · quiz
- 5 Separation axioms and metrizability 27 min · quiz
- 6 Connectedness and path-connectedness 24 min · quiz
- 7 Compactness and the Heine–Borel theorem 26 min · quiz
- 8 Tychonoff’s theorem, compactification and local compactness 26 min · quiz
- 9 Homotopy and the fundamental group 27 min · quiz
- 10 The fundamental group of the circle and its applications 28 min · quiz
- 11 Covering spaces and the Seifert–van Kampen theorem 29 min · quiz
- 12 Capstone: surfaces, Euler characteristic and the classification theorem 27 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
Shape without distance: how continuity depends only on open sets, topological spaces defined by open sets with bases, subbases, closures, interiors and subspaces, and examples from metric, discrete, cofinite, Sierpiński and lower limit topologies, continuous maps, the pasting lemma and homeomorphisms such as stereographic projection, product and quotient topologies with their universal properties and the torus, Möbius strip, Klein bottle and projective plane glued from a square, the separation axioms with Urysohn's lemma, the Urysohn metrization theorem and the non-normal Sorgenfrey plane, connectedness of intervals, path-connectedness and the topologist's sine curve, compactness with the tube lemma, the Heine–Borel theorem and continuous bijections from compact to Hausdorff spaces, Tychonoff's theorem proved with maximal families and Zorn's lemma, local compactness and the one-point compactification, homotopy and the fundamental group, the fundamental group of the circle by path lifting with the Brouwer fixed point theorem, the fundamental theorem of algebra and the Borsuk–Ulam theorem in dimension 2, covering spaces and the Seifert–van Kampen theorem, and a capstone classifying closed surfaces by orientability and Euler characteristic, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 4.