In my previous comment on boundaries in object universes and venn diagrams, and always when I’m being careful about their mathematical senses, the definitions of “topology” and “boundary” I have in mind can be found in any standard textbook.  Here are links to basic definitions from J.L. Kelley, a veritable classic and my own first brush with the subject.

Excerpts from John L. Kelley, General Topology, Van Nostrand Reinhold, New York, NY, 1955

• Mathematical Notes • Topology
• Definition of Topology • Note 1
• Definition of Interior and Boundary • Note 10 • Note 11 • Note 12 • Note 13