Theorem 5.1 Existence and uniqueness of the complex numbers

Proposition 5.6 Convergent sequences in metric spaces have unique limits

Proposition 5.7 Condition for continuity in terms of sequences

Proposition 5.8 A test for convergence in a metric space

Proposition 5.9 The projection and embedding maps in Rⁿ are continuous

Proposition 5.10 Arithmetic with continuous maps

Proposition 5.11 The composition of continuous maps is continuous

Proposition 5.12 The complement of an open set is closed

Proposition 5.13 Open neighborhoods are open sets

Proposition 5.14 Criterion for continuity in terms of open sets

Corollary 5.15 Criterion for continuity in terms of closed sets

Proposition 5.16 Finite subsets in metric spaces are closed

Proposition 5.17 A Union of arbitrary open sets is open

Proposition 5.18 An Intersection of arbitrary closed sets is closed

Proposition 5.19 The union of finitely many closed sets is closed

Proposition 5.20 The intersection of finitely many open sets is open

Theorem 5.21 A subset of R is connected iff it is an interval

Theorem 5.22 Continuous functions map connected sets to connected sets

Theorem 5.23 Intermediate Value Theorem