A characterization of the uniform closure of the set of homeomorphisms of a compact totally disconnected metric space into itself

Abstract:

The limit index $\lambda (x)$ of a point $x$ in a compact metric space is defined. (Roughly: Isolated points have index 0, limit points have index 1, limit points of limit points have index 2, and so forth.) Then the following theorem is proved. Theorem 1. Let $E$ be a compact, totally disconnected metric space. Then the uniform closure of the set of homeomorphisms of $E$ into itself is the set ${C_\lambda }$ of continuous functions $f$ from $E$ to $E$ satisfying (1) $\lambda (x) \leqslant \lambda (f(x))\;for\;all\;x \in E$, and (2) if $y$ is not a condensation point of $E$, then ${f^{ - 1}}(y)$ contains at most one $x$ such that $\lambda (x) = \lambda (y)$. Further, the set of homeomorphisms of $E$ into $E$ is a dense ${G_\delta }$ subset of the complete metric space ${C_\lambda }$.