By Henry Bradford Smith

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58X If A is scattered, then (AnCl( 0))= 0. Comment There are no theorems of logic in a scattered BASIC TOPOLOGICAL PROPERTIES 41 set. 59X 0 is scattered. 60X 0 is closed if and only if some scattered set is closed. Comment There are theorems of logic if and only if every theory has nonempty, pointwise redundant subsets. 61X If each point is closure equivalent to some other point, then no nonempty, scattered set is open. Comment Suppose that each sentence in the language L is deductively equivalent to some other sentence ofL.

29 If some member of S is contradictory, then the boundary of each consistent subset A of S is just CI(A). Proof Suppose {x} is dense and A is consistent. 27, IntiA)« 0. So Bdry(A)=CI(A)\ 0 . 23 are reminiscent of familiar theses from modal logic. We conclude this section by exploring the relationship between interior operators and the necessity operator of the modal logic 84. (Cf. McKinsey, pp. 128-134. ) If ACS and WC:P(S), then we say that Wr A if and only if n W C A. We let D A=Int(A). 22 implies that {DA}r A (the modal principle of necessity elimination).

7 xECl(A\ {xl) if and only if xE A ' . 6, xE Cl(A\ {x]) if and only if there are members of A\{x} in each neighborhood ofx. '). 8 [x] is independent if and only ifx¢Cl( 0). 7, {x}"=Cl( 0) (since {x}\ {x}= 0). So {x] is independent if and only if (Ixlf) C1( 0 )) = 0 . But ({x}nCl( 0))= 0 if and only ifx¢Cl( 0). Comment An axiom system consisting of a single sentence is independent just in case that sentence is not a theorem of logic. Unfortunately, this somewhat undermines the notion of independence.