Mathematics / core
Real analysis in metric spaces
Make convergence and continuity rigorous through completeness, compactness, fixed points, and metric-space counterexamples.
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Inside this module
Module 01 core
Real analysis in metric spaces
Make convergence and continuity rigorous through completeness, compactness, fixed points, and metric-space counterexamples.
- 01
Metrics, open balls, and neighborhoods
Turn a distance rule into a precise language of local behavior and compare different geometries on the same set.
26 min - 02
Convergence and closed sets
Read convergence as an eventual neighborhood condition and characterize closed sets by the limits of their sequences.
27 min - 03
Cauchy sequences and completeness
Separate tail agreement from the existence of a limit and detect the missing points of an incomplete space.
29 min - 04
Completions and dense subsets
Construct missing limits from Cauchy sequences and understand density as approximation at every scale.
31 min - 05
Continuity and uniform continuity
Control output errors locally or across an entire domain and learn how compactness supplies a uniform tolerance.
29 min - 06
Sequential compactness and total boundedness
Separate finite-resolution coverage from ordinary boundedness and derive convergent subsequences through nested selections.
32 min - 07
Compactness and extreme value theorems
Use finite subcovers to turn local information into global guarantees and prove that continuous objectives attain extrema.
32 min - 08
Connectedness and intermediate values
Express the absence of a separation and use continuous images to explain why intermediate values cannot be skipped.
30 min - 09
Contraction mappings and fixed points
Turn a shrinking distance estimate into a unique fixed point, a convergent iteration, and a usable error certificate.
34 min - 10
Baire category and complete spaces
Use nested complete-space constructions to show that countably many nowhere dense obstructions cannot exhaust an open region.
35 min