Rings of Continuous Functions by Leonard Gillman

By Leonard Gillman

Edited by way of Oscar Zariski, S. S. Chern, And L. Nirenberg.

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Examples of z-filters on R that converge to 0 are provided by the families of all zero-sets Z in R satisfying the respective conditions: (i): Z is a neighborhood of 0; (ii) : there exists E > 0 such that Z contains the interval [0, E]; (iii): there exists E > 0 such that Z contains [ - E, 0] ; (iv): 1jn E Z for all but finitely many n EN; (v): 0 E Z. 18(b )), the last of these is a z-ultrafilter, and it is the only z-ultrafilter that converges to O. 'F converges to p. Let rff denote the z-filter of all zero-set-neighborhoods of p.

For each m ~ No, each ring on R, N, or N* contains a function having exactly 2m square roots. If a member of C(Q) has more than one square root, it has c of them. 2. C(R) has just two idempotents, C(N*) has exactly No, and C(Q) and C(N) have c. 3. Every nonzero idempotent in C(Q) is a sum of two nonzero idempotents. In C(N), and in C(N*), some, but not all idempotents have this property. 4. Except for the obvious identity C(N*) = C*(N*), no two of the rings in question are isomorphic. 5. Each of C(Q) and C(N) is isomorphic with a direct sum of two copies of itself.

2) that every closed ideal is an intersection of maximal ideals. ) 6. The topology of uniform convergence can also be defined on C, the 2N IDEALS AND Z-FILTERS 35 neighborhood system at g being as described in 2. However, C will not be either a topological ring or a topological vector space unless X is pseudocompact. 2N. THE m-TOPOLOGY ON C. The m-topology is defined on C(X) by taking as a base for the neighborhood system at g all sets of the form {f E C: Ig - II ~ u}, where u is a positive unit of C.

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