Unit · year 3
MU-301 · Metric & Topological Spaces
Threads space25 lectures5 theorems
Continuity, compactness and completeness without coordinates.
PREREQUISITES
Lectures
| L01 | Metric Spaces: Axioms and Examples — |
| L02 | Open Balls, Open Sets, and Closed Sets — |
| L03 | Convergence in Metric Spaces — |
| L04 | Continuity in Metric Spaces — |
| L05 | Topological Spaces: The Axioms — |
| L06 | Bases, Subbases, and Generated Topologies — |
| L07 | Closure, Interior, and Boundary — |
| L08 | Continuity via Preimages |
| L09 | Homeomorphisms and Topological Invariants |
| L10 | Subspace, Product, and Quotient Topologies — |
| L11 | Separation Axioms: Hausdorff and Beyond — |
| L12 | Compactness: The Open Cover Definition |
| L13 | Compactness in Metric Spaces |
| L14 | Sequential Compactness and Total Boundedness |
| L15 | Tychonoff's Theorem |
| L16 | Connectedness and Path-Connectedness — |
| L17 | Components and Local Connectedness — |
| L18 | Completeness and Cauchy Sequences — |
| L19 | Completion of a Metric Space — |
| L20 | The Banach Fixed-Point Theorem |
| L21 | Applications: ODEs and Numerical Schemes |
| L22 | The Baire Category Theorem |
| L23 | Consequences of Baire Category |
| L24 | Compactness in Function Spaces: Arzelà –Ascoli |
| L25 | Synthesis: What Topology Keeps and What It Forgets |
Theorems in this unit
T-079
Continuity via preimages
A map is continuous iff preimages of open sets are open.
T-080
Characterisations of compactness
Open-cover and sequential compactness coincide in metric spaces.
T-081
The Banach fixed-point theorem
A contraction on a complete space has a unique fixed point.
T-082
The Baire category theorem
A complete metric space is not a countable union of nowhere-dense sets.
T-083
Tychonoff's theorem
An arbitrary product of compact spaces is compact.