By Norman M. Martin

This booklet examines an summary mathematical concept, putting precise emphasis on effects appropriate to formal good judgment. If a thought is principally summary, it could discover a common domestic inside of numerous of the extra frequent branches of arithmetic. this is often the case with the idea of closure areas. it'd be thought of a part of topology, lattice concept, common algebra or, without doubt, one of many different branches of arithmetic besides. In our improvement we now have handled it, conceptually and methodologically, as a part of topology, partially simply because we first notion ofthe simple constitution concerned (closure space), as a generalization of Frechet's suggestion V-space. V-spaces were utilized in a few advancements of basic topology as a generalization of topological house. certainly, whilst within the early '50s, certainly one of us began brooding about closure areas, we idea ofit because the generalization of Frechet V house which comes from no longer requiring the null set to be CLOSURE areas ANDLOGIC XlI closed(as it really is in V-spaces). This generalization has an severe virtue in reference to software to good judgment, because the most vital closure concept in good judgment, deductive closure, often doesn't generate a V-space, because the closure of the null set ordinarily comprises the "logical truths" of the common sense being examined.

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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.