Three extended real interval systems are defined and distinguished by their implementation complexity and result sharpness. The three systems are closed with respect to interval arithmetic and the enclosure of functions and relations, notwithstanding domain restrictions or the presence of singularities. 1 Overview Section 2 introduces the problem of defining closed interval systems. In Section 3, real and extended points and intervals are defined. In Section 4, the empty and entire intervals are used to close the extended interval system. Section ?? shows how incorrect conclusions have been reached about the result of certain interval arithmetic operator-operand combinations. The author is grateful to Professor Arnold Neumaier for originally raising this issue. Section 9 describes how to legitimately use IEEE floating-point arithmetic to obtain the sharp results described in Section ??. Section 6 generalizes extended interval arithmetic to define interval enclosures of functions, with..