Partitions of unity
  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. 
Jerzy Dydak