Internally \(\mathcal K\)-like spaces and internal inverse limits (Q2435279)

The main result of this paper is: { Theorem 2.} Let \(X\) be a compact metric space, \((Y_n)\) be a sequence of closed subsets of \(X\) with \(\mathrm{Lim}Y_n=Y\subset X\), and \(\mathcal{F}\) be a collection of maps. Suppose that for each \(\epsilon>0\), there is an \(N(\epsilon)\in\mathbb N\) such that for each \(n>N(\epsilon)\), there exists a uniformly equicontinuous sequence \((f_n^m)\) of maps \(f_n^m:Y_m\to Y_n\) in \(\mathcal{F}\), \(m>n\), with \(\tilde d(f_n^m)<\epsilon\). Then there are a subsequence \((Y_{n_k})\) and maps \(g_k:Y_{n_{k+1}}\to Y_{n_k}\) in \(\mathcal{F}\) such that the inverse sequence \((Y_{n_k},g_k)\) converges exactly in \(X\) to \(Y\). If, additionally, \(Y_n\subset Y\) for each \(n\), then \((Y_{n_k},g_k)\) is an internal inverse limit structure on \(Y\). Although not in the authors' statement of this theorem, we believe that they meant to require that \(X\) be compact and metric. To give an idea of the meaning of this result, we define some of the terms. First, by \(\mathrm{Lim}Y_n\) is meant the limit of the sequence \((Y_n)\) in the hyperspace \(2^X\). The collection \(\{f_n^m\,|\,m>n\}\) of maps \(f_n^m:Y_m\to Y_n\) is uniformly equicontinuous requires that for all \(\epsilon>0\), there exists \(\delta>0\) such that \(d(f_n^m(x),f_n^m(y))<\epsilon\) for all \(m\) and for all \(x\), \(y\in Y_m\). The requirement that \(\tilde d(f_n^m)< \epsilon\) is that \(\sup\{d(x,f_n^m(x))\,|\,x\in Y_m\}<\epsilon\). The more complicated notions of ``converges exactly'' and ``internal limit structure'' are given on p. 236; we will not repeat them here.