  The paper contains an exposition of part of topology using partitions of
unity. The main idea is to create variants of the Tietze Extension Theorem and
use them to derive classical theorems. This idea leads to a new result
generalizing major results on paracompactness (Stone Theorem and Tamano
Theorem), a result which serves as a connection to Ascoli Theorem. A new
calculus of partitions of unity is introduced with applications to dimension
theory and metric simplicial complexes. The geometric interpretation of this
calculus is the barycentric subdivision of simplicial complexes. Also, joins of
partitions of unity are often used; they are an algebraic version of joins of
simplicial complexes.
