Limit theorem for inverse sequences of metric spaces in extension theory (Q1306246)

Let \(X\) be a topological space and \(L\) be some fixed space. We say that \(X\) has the extension property with respect to \(L\) if, given any closed subset \(A\) of \(X\) and a continuous map \(f:A\to L\), there is an extension \(F:X \to L\) of \(f\). We abbreviate this as \(X\tau L\). In this paper the authors prove the following result: Suppose \(X\) is the inverse limit of a sequence \(\{X_i\}\) of metric spaces and \(K\) is a simplicial complex. Then \(X_i\tau |K|\) for all \(i\in \mathbb{N}\) implies that \(X\tau |K|K\). The result is quite interesting and at once yields the following two corollaries, the first of which is already known: (i) (Nagami) If \(\dim X_i \leq n\), then \(\dim X \leq n\). (ii) If \(\dim _GX_i\leq n\), then \(\dim _GX\leq n\). Here dim stands for the covering dimension and \(\dim _GX\) denotes the cohomological dimension of \(X\) with respect to the abelian group \(G\); \(|K|\) denotes the simplicial complex \(K\) with the weak topology.