By Conforto F.

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If T is a Compact Hausdorff space then the maximal ideals of C(T,R) are all fixed. Proof We show that there are no free ideals in C(T,R) when T is compact: we show that if M is free then M contains a unit. If M is free then for each t xt(t) # 0. F T there is some xt 6 M such that Since each xt is continuous there must be open neighborhoods Ut ALGEBRAS OF CONTINUOUS FUNCTIONS 1. 18 for each t on which x does not vanish. ,Ut say, must cover T. It is now easy to see that the n function y = x2 + + x2 tn *** tl

4-1 enables us to characterize compactness externally, namely: The completely regular Hausdorff space T is compact if and only if each maximal ideal in C(T,R) is fixed. To see the sufficiency of the condition, we need only note that if T is not compact, then for any p C BT-T, M P is not fixed. The algebra C(T,C) of complex-valued continuous functions does not differ markedly from C(T,R) if one takes preservation of the results of this chapter as the datum. the C(T,C)-completion. The C(T,R)-completion of T yields BT; so does The maximal ideals of C ( T , C ) are in 1-1 corre- spondence with the maximal ideals of C(T,R) under the mapping M + re M + i re M where re M denotes the collection of real parts re x of functions x in M.

We prove countable compactness by showing each sequence to have a cluster point. If (t ) is a sequence from [O,n), it can be rearranged so that it is increasing. The rearranged sequence then has its supremum, t say, as a limit. e. t € [O,n) -and is a cluster point of the original sequence. 5-3) that [O,a) is not replete. Later conditions under which spaces C(T,R,c) of continuous functions with compact-open topology are barreled and bornological are obtained: 28 1. (Th. 5-1) functions ALGEBRAS OF CONTINUOUS F U N C T I O N S C(T,R,c) is barreled if and only if unbounded continuous XI exist on each closed noncompact subset E of T.