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John;The open interval (a1, a2) is the interior of the closed interval [a1, a2]. So I have no problem with calling an interval [b1, b2] that is in (a1, a2) interior to [a1, a2].
The comparison called 'interior' in Motion 13.04 is frequently used in verified computing. For instance, it is frequently used establishing existence, uniqueness, and eclosure in case of systems of linear equations (sufficient condition).
If you do not like the denotation, please suggest another name. Best regards Ulrich Am 19.09.2010 09:10, schrieb John Pryce:
Nate On 18 Sep 2010, at 23:15, Nate Hayes wrote:I speak against this. Ulrich's interior is better. Note that the topological interior, i.e., "proper subset," is already expressed efficiently in terms of Ulrich's relations for intervals A,B: ( A \subset B ) and not ( A == B )Doesn't that make [2,3] interior to [1,3]?I don't see Entire should be interior to Entire.Well, it seems weird to me too, but there it is. You're an expert on quantified statements. Isn't it inescapable from the definition "B is a neighbourhood of each a in A" (eqn1)? The thing about definitions grounded in standard theory (and this theory has been around for roughly 100 years) is that, compared with ad-hoc definitions, you KNOW they can't lead to inconsistencies -- assuming math itself is consistent. John
-- Karlsruher Institut für Technologie (KIT) Institut für Angewandte und Numerische Mathematik (IANM2) D-76128 Karlsruhe, Germany Prof. Ulrich Kulisch Telefon: +49 721 608-2680 Fax: +49 721 608-6679 E-Mail: ulrich.kulisch@xxxxxxx www.kit.edu www.math.kit.edu/ianm2/~kulisch/ KIT - Universität des Landes Baden-Württemberg und nationales Großforschungszentrum in der Helmholtz-Gemeinschaft