In this article, we present a closely related approach to the study of N(X ) , S (X ) , and the spectral theorem for unbounded self-adJoint operators. We begin in Sec- tion 2 with a theorem (Theorem 2.1) on continuous extensions from open dense subsets of extremely disconnected spaces (see also [3, p. 96]). Theorem 2.1 leads to a substantial simplification of the proof that N(X) is an algebra, and it plays a key role in our development. We continue, in Section 3, with a discussion on the spectral analysis of a function in S(X), and we give an altemative proof of the fact that S(X) is a boundediy complete lattice. In Section 4 we prove the spec- tral theorem and characterizations of the spectrum and the spectral projections for unbounded self-adJoint operators .
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