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I'm looking for a reference which proves some basic statements about basically disconnected spaces. In particular:

The fact that if $X$ is compact Haussdorf basically disconnected that then $C(X)$ is $\sigma$-Dedekind complete (and vice versa) or equivalently that a commutative C$^*$ algebra is $\sigma$-Dedekind complete if and only if its Gelfand spectrum is basically disconnected.

And the related fact that the linear span of the characteristic functions of clopen sets lies dense in $C(X)$.

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