Since X is closed and a

proper subset of T, we may choose a closed interval I [subset or equal to] X of maximal length.

If X contains a nonempty

proper subset A which is both [[tau].sub.1]-[delta] semiopen and [[tau].sub.2]-[delta] semiclosed.

Sufficiency: Suppose A is non-empty

proper subset of X such that it is [alpha]-open as well [alpha]-closed.

We only need to prove that X is not a

proper subset of U.

Given (B, [disjunction], [and]) is a Boolean-near-ring whose

proper subset (A, [disjunction], [and]) is a maximal set with uni-element in an associate ring R, and if the maximal set A is also a subset of B.

The only

proper subset in this situation is the empty set.

(1) above): (9) the members of each collection include those of the next as the

proper subset (with soldier Brown representing a limiting case of a subset, though not of a collection, since the latter has two members as its limiting case).

We now show that sets of FDs that are monodependent and optimum are closed under the

proper subset operation.

Provisionally defining a minimal sufficient condition as one which has no sufficient condition as a

proper subset, let us see how this works on a couple of examples.

Let [Omega] be a connected open set in C and E be a relatively closed

proper subset of [Omega].

The class of short stories is a

proper subset of the class of short texts, for instance, or the class of narrative texts, the class of written texts, the class of written narrative texts, and so on.

(2) Let [{[e.sub.j]}.sub.j[member of]J] be a

proper subset of E whose cardinal is larger than continuum and then consider [O.sub.s] = [U.sub.j[member of]j]{x [member of] B(H) : [parallel][xe.sub.j][parallel] < 1} which forms an open set in the strong operator topology.