Let \(B\) be a commutative semisimple Banach algebra and let \(\Delta_B\) be its maximal ideal space. The question of relating topological invariants of \(\Delta_B\) to algebraic invariants of \(B\) has attained much interest since the early days of commutative Banach algebras, starting with Shilov's idempotent theorem which identifies \(H^0(\Delta_B, \mathbf Z)\), the 0'th Čech cohomology with integral coefficients, and the additive group generated by idempotents of \(B\). Subsequently groups of higher order have been determined. \(H^1(\Delta_B, \mathbf Z)\) is the invertible group of \(B\) modulo elements with a logarithm (Arens and Royden), \(H^2(\Delta_B, \mathbf Z)\) is the Picard group of \(B\) (Forster), and \(H^3(\Delta_B, \mathbf Z)\) is the bigger Brauer group of \(B\) (Taylor). Likewise the fundamental group \(\pi_1(\Delta_B)\) has been the subject of investigations, i.e. the relation between covering spaces of \(\Delta_B\) and algebraic invariants of \(B\). The paper under review is in this line. The following theorem is proved \textbf{Theorem} For every locally connected compact Hausdorff space \(X\) the category of regular covering spaces of \(X\) and the category of Galois extensions of \(C(X)\) are equivalent. Here \(C(X)\) denotes the ring of continuous complex valued functions on \(X\) and Galois extensions is meant in the sense that the author defines, that is, extensions are allowed to be non-unital. This result extends a result by Childs and Wajnryb who both have shown that there is a category equivalence between the finite-fibered covering spaces of \(X\) and Galois extensions in the classical sense. On the route to the theorem above the author develops the necessary theory for non-unital extensions.