The regular open subsets of a topological space form Boolean algebra, where the join two sets is interior closure their union. A content finitely additive measure on this or one its subalgebras. We develop theory integration for such contents. then explain relationship between contents, residual charges, and Borel measures. show that can be represented by normal measure, augmented with liminal ...

[1] B. Banaschewski, Über nulldimensionale Räume, Math. Nache 13 (1955) 129-140. [2] F. Borceux and J. Janelidze, Galois Theories, Cambridge University Press (2001). [3] M. Dias and M. Sobral, Descent for Priestley Spaces, Appl. Categor. Struct 14 (2006) 229-241. [4] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge Mathematical Texbooks (1990). [5] R. Engelking and...

In this note the Stone-Cech compactification is used to produce short proofs of two theorems on the structure of free topological groups. The first is: The free topological group on any Tychonoff space X contains, as a closed subspace, a homeomorphic copy of the product space X". This is a generalization of a result of B. V. S. Thomas. The second theorem proved is C. Joiner's, Fundamental Lemma.