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PROOF. For each point P of D, there exists a region gp of Gi containing P such that gp is a subset of D. By Axiom 2, there exists a connected domain dp containing P which is a subset of gp. Let Ui denote the collection of all domains dp for each point P of D. The point set M is closed and compact, and hence, by Theorem 22 of Chapter I, it is covered by a finite subcollection W\ of U\. By Theorem 77 of Chapter I, for each pair of domains x and y of W\ there exists a simple chain xy whose links are domains of Ui and whose first and last links are x and y respectively. Let Vi denote the collection of all domains v such that for some two domains x and y of Wu v is a link of the chain xy. The sum of all the domains of the finite collection Vi is a connected domain Du Similarly, there exists a finite collection Vi of connected domains such that if v is any domain of V2, then v is a subset of some region of Gi and of some domain of Vu and such that the sum of the domains of Vi is a connected domain Di. This process can be continued. Thus there exists an infinite sequence Vu Vi, V3, • • • such that, for each n, (1) Vn+i is a finite collection of connected domains such that if v is any one of them then v is a subset of some region of Gn+i and of some domain of Vn and of Dy and (2) the sum of all the domains of Vn is a connected domain Dr containing M. By Theorems 79 and 80 of Chapter I, the set of all points common to all the sets of the sequence Du Di, Dz, • • • is a compact continuum, and it contains M and lies in D. A modification of this argument proves this theorem for a space which satisfies Axioms 0 and 1 and is locally arcwise connected.

A theorem concerning closed and compact point sets which lie in connected domains