Corollary 2

Let Λ be a set and let $X\subset \mathbf{C}^{\Lambda}$ be a compact set. If X is connected, then its polynomially convex envelope is connected.

If X is connected, the only idempotents of $\mathscr{C}(X)$, and hence of P(X), are 0 and 1 (cor. of prop. I, p. 79). Consequently, the space $\mathsf{X}(P(X))$ is connected (loc. cit.); now this set is homeomorphic to the polynomially convex envelope of X (cor. 1, a)).