# Math 104 Knowledge Map

This page is the ordinary reading view of the graph. It favors useful mathematical relationships over exhaustive cataloging. Precise records are in `nodes.csv` and `edges.csv`.

## Course spine

```text
Ordered fields and completeness
        ↓
Metric spaces and topology
        ↓
Compactness and connectedness
        ↓
Sequences, completeness, and series
        ↓
Limits and continuity of functions
        ↓
Differentiation
        ↓
Riemann–Stieltjes integration
        ↓
Sequences and series of functions
        ↓
Uniform convergence and equicontinuity
        ↓
Arzelà–Ascoli and approximation theorems
```

This is a structural overview, not a claim that each topic has only one prerequisite.

## Week 1: ordered fields and completeness

```text
Ordered set
   ↓
Ordered field
   ↓
Least-upper-bound property / completeness
   ├── Archimedean property
   ├── Density of Q in R
   └── Existence of positive nth roots

Euclidean-space structure
   └── Cauchy–Schwarz inequality
          └── Triangle inequality
```

### What to be able to do

- State upper bound, supremum, ordered field, and least-upper-bound property precisely.
- Explain in Chinese what completeness adds that the ordered-field axioms alone do not provide.
- Reproduce the contradiction structure behind the Archimedean property.
- Explain how density uses the Archimedean property.
- Separate the formal statement of an nth-root result from the construction used in its proof.
- Recognize Cauchy–Schwarz as the key estimate behind the Euclidean triangle inequality.

### Source boundary

The detailed Week 1 nodes below were checked against the current course handout and the maintained Week 1 note. The higher-level course spine is structurally inferred from syllabus order and standard prerequisite flow; it should not be attributed verbatim to Rudin or Ross.

## How to extend this map

Add a node when it is a durable definition, theorem, counterexample, proof technique, or exercise method. Add an edge only when the relationship helps answer one of these questions:

- What must I know before attempting this proof?
- Where is this result used later?
- Which assumption prevents a standard counterexample?
- Which exercise reuses the same proof idea?
- Is this relationship stated by the source or inferred for learning?

Do not add a node merely because a term appears once in a note.