Table of Contents

Two Setups in TDA
Sensing continuous Functions
Point Clouds of Shapes
Distances on Functions
Absolute Distance of Functions
Relative Distance of Functions
Lower Bounds
Further References
Reeb Graphs
Join Trees
Interlude on Precosheaves
Monoidal Posets for 1D-Interleavings
Interleaving Reeb Graphs
Interleaving Precosheaves on D
Interleavings in D-Categories
Properties of Interleavings
Homomorphisms of D-Categories
Positive Persistence-Enhancements
Complete Persistence-Enhancements
Some Equivalences
Equivalence to descending Precosheaf
A second type of Interleavings
A negative Enhancement for Join Trees
Equality of Interleaving Distances
Relation to the Join Precosheaf
Constructible Spaces over the Reals
Graphs over the Reals
A Skeleton for the Epigraph