Topology
Year 3 · 5 sections · 62 concepts · 96 practice questions
What's covered
free1 · Topological spaces16 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Topological spaces
- Definition 1.2 · Bases
- Proposition 1.3 · The topology generated by a basis
- Example 1.4 · Standard topologies
- Definition 1.5 · Closed sets, closure and interior
- Proposition 1.6 · Characterising the closure
- Definition 1.7 · Continuous maps
- Proposition 1.8 · Equivalent forms of continuity
- Definition 1.9 · Homeomorphisms
- Definition 1.10 · The subspace topology
- Definition 1.11 · The product topology
- Proposition 1.12 · Universal property of the product
- Definition 1.13 · Quotient spaces
- Example 1.14 · Gluing spaces
- Check 1.15 · Check: proving two spaces are (not) homeomorphic
2 · Separation and countability10 concepts
- Overview 2 · The plan of this section
- Definition 2.1 · Hausdorff spaces
- Proposition 2.2 · Limits are unique in Hausdorff spaces
- Proposition 2.3 · Subspaces and products of Hausdorff spaces
- Definition 2.4 · Regular and normal spaces
- Proposition 2.5 · Metric spaces are normal
- Theorem 2.6 · Urysohn's lemma
- Definition 2.7 · First and second countable spaces
- Proposition 2.8 · Sequences suffice in first-countable spaces
- Check 2.9 · Check: using separation and countability
3 · Compactness12 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Compactness
- Proposition 3.2 · Closed subsets of compact sets
- Theorem 3.3 · Continuous images of compact sets
- Proposition 3.4 · Compact subsets of Hausdorff spaces are closed
- Theorem 3.5 · Continuous bijections from compact to Hausdorff spaces
- Theorem 3.6 · Finite products of compact spaces are compact
- Corollary 3.7 · Heine–Borel theorem
- Lemma 3.8 · The Lebesgue number lemma
- Theorem 3.9 · Compactness in metric spaces
- Theorem 3.10 · The Baire category theorem
- Check 3.11 · Check: compactness arguments
4 · Connectedness12 concepts
- Overview 4 · The plan of this section
- Definition 4.1 · Connected spaces
- Proposition 4.2 · Intervals are connected
- Theorem 4.3 · Continuous images of connected spaces
- Proposition 4.4 · Unions and products of connected spaces
- Definition 4.5 · Connected components
- Definition 4.6 · Path-connected spaces
- Proposition 4.7 · Path-connected spaces are connected
- Example 4.8 · The topologist's sine curve
- Proposition 4.9 · Open connected subsets of ℝⁿ are path-connected
- Example 4.10 · Telling spaces apart by connectedness
- Check 4.11 · Check: connectedness arguments
5 · The fundamental group12 concepts
- Overview 5 · The plan of this section
- Definition 5.1 · Homotopy
- Definition 5.2 · The fundamental group
- Theorem 5.3 · The fundamental group is a group
- Proposition 5.4 · Changing the basepoint
- Proposition 5.5 · Induced homomorphisms
- Definition 5.6 · Simply connected spaces
- Example 5.7 · Convex and star-shaped sets
- Theorem 5.8 · The fundamental group of the circle
- Corollary 5.9 · No retraction of the disc onto its boundary
- Corollary 5.10 · Brouwer's fixed point theorem in dimension two
- Check 5.11 · Check: computing and using fundamental groups
Try it: a free card from this module
This is exactly what studying looks like inside UniMaths.
Proposition 1.3 · The topology generated by a basis
open ⇔ every point has a basis set around it inside .
Proof
Learn all 62 concepts like this
Cards, step-by-step proofs, and 96 questions scheduled so you review right before you forget.