Unit · year 3
MU-301 · Metric & Topological Spaces
Threads space5 theorems
Continuity, compactness and completeness without coordinates.
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.