On Inverse Limits of Metric Spaces

Abstract

Inverse limit spaces have been a topic studied in various fields of mathematics such as Algebra, Measure Theory, and Topology. Here, we present a theorem that can be summarized as a game in which a given compact metric space $X$ is expressed as an inverse limit built step-by-step by two players. In the $i$-th step of the game, the first player gives an $\epsilon_i > 0$ and the second player gives a complete space $Y_i$ and two maps, $f_i: X \rightarrow Y_i$ and $g_{i-1}: Y_i \rightarrow Y_{i-1}$ with the conditions that $\textnormal{dist}(f_{i-1},g_{i-1}\circ f_i) < \epsilon_i$, and $f_i$ does not mend any two points of $X$ with distance greater than some $\eta_i$ where $\displaystyle{\lim_{i\rightarrow \infty}}\eta_i = 0$. We prove that the first player can cause the sequence $(g_i \circ \cdots \circ g_{j-1} \circ f_j)_{j=i}^\infty$ to converge uniformly to a map $\tilde{f}_i: X \rightarrow Y_i$ for each $i$, and that the map $\tilde{f}$ induced by $\tilde{f}_0, \tilde{f}_1, \ldots$ is a homeomorphism of $X$ onto $\varprojlim\{\tilde{f}_i(X), g_i\}_{i=0}^\infty$. Classic theorems by Anderson-Choquet, Marde\v si\' c-Segal, and Morton Brown can be reproved by using elements of this game.