Lemma 1

For every proper partial mapping $f$ of X into Y, with domain of definition U, let us denote by $\widetilde{f}$ the mapping of $X'$ into $Y'$ defined by $\widetilde{f}(x) =f(x)$ if $x\in U$ and $\widetilde{f}(x) =\omega_Y$ if $x \notin U$; it is continuous.

The mapping $f\mapsto \widetilde{f}$ is a bijection between the set of proper partial mappings $f$ of X into Y and the set of continuous mappings $g$ of $X'$ into $Y'$ such that $g(\omega_X) =\omega_Y$.

Let $f$ be a proper partial mapping of X into Y, and let U be its domain. We shall prove that the mapping $\widetilde{f}$ is continuous. It is continuous at every point of U, since U is open in $X'$. We shall prove that it is also continuous at every point $x$ of $X'-$ U; then $\widetilde{f}(x) =\omega_Y$. Let V be an open neighbourhood of $\omega_Y$ in $Y'$; we shall prove that $\widetilde{f}^{-1}(V)$ is a neighbourhood of $x$. By definition of the topological space $Y'$, we may suppose that V is of the form $Y'-$ K, where K is a compact subset of Y. Since $f$ defines a proper mapping of U into Y, the set $f^{-1}(K)$ is compact in U (TG, I, p. 77, prop. 6), hence in $X'$. In particular it is a closed subset of $X'$, and $\widetilde{f}^{-1}(V) =$ $X'-f^{-1}(K)$ is an open subset of $X'$, and is therefore a neighbourhood of $x$.

Conversely, let $g: X'\rightarrow Y'$ be a continuous mapping such that $g(\omega_X) =\omega_Y$, and let $\Gamma_g\subset X'\times Y'$ be its graph. The set U = X $-\overset{-1}{g}(\omega_Y)$ is open in X. The correspondence $f= (\Gamma_g\cap (U\times Y),X,Y)$ is a proper partial mapping of X into Y (TG, I, p. 77, prop. 7) such that $\widetilde{f}=g$, and it is the only one.

We shall identify the proper partial mappings of X into Y with the continuous mappings of $X'$ into $Y'$ which map $\omega_X$ onto $\omega_Y$. In particular, the proper mappings of X into Y are the proper partial mappings with domain X; they are identified with the continuous mappings $f$ of $X'$ into $Y'$ such that that $\overset{-1}{f}(\omega_Y) =\{\omega_X\}$. If X is compact, these are simply the continuous mappings of X into Y.

Let A be a commutative complex Banach algebra. Recall that $\mathsf{X}'(A)$ is identified with the compact space obtained from $\mathsf{X}(A)$ by adjoining a point at infinity (I, p. 29, corollaire).