An extension of Michael's selection theorem (Q540778)

Let \(X\) be a paracompact zero-dimensional space (in the sense of the covering dimension) and \((Y,d)\) a complete metric space. Let \(2^Y\) denote the set of all nonempty subsets of \(Y\). For a multifunction \(T : X \to 2^Y\), a function \(f: X \to Y\) is called a selection of \(T\) if \(f(x) \in T(x)\) for each \(x \in X\). A multifunction \(T : X \to 2^Y\) is said to be lower semicontinuous (l.s.c.)\ if for every \(x \in X\) and every open subset \(G\) of \(Y\) with \(G\cap T(x) \neq \emptyset\), there exists a neighborhood \(V_x\) of \(x\) such that \(G \cap T(z) \neq \emptyset\) for each \(z \in V_x\). A multifunction \(T : X \to 2^Y\) is said to be almost lower semicontinuous (a.l.s.c.)\ if for every \(x \in X\) and \(\epsilon>0\), there exists a neighborhood \(V_x\) of \(x\) such that \(\bigcap_{z \in V_x} B_\epsilon (T(z)) \neq \emptyset\), where \(B_\epsilon (T(x)) = \{ y \in Y : d(y,T(x))<\epsilon\}\). \textit{E. Michael} [Am.\ Math.\ Mon.\ 63, 233--238 (1956; Zbl 0070.39502)] proved that every l.s.c.\ closed-valued multifunction \(T: X \to 2^Y\) admits a continuous selection. This theorem cannot be extended to a.l.s.c.\ multifunctions. In general, an a.l.s.c.\ closed-valued multifunction \(T: X\to 2^Y\) need not admit a continuous selection. In this paper, the authors give a sufficient condition for a.l.s.c.\ compact-valued multifunctions \(T: X \to 2^Y\) to admit continuous selections. A multifunction \(T: X \to 2^Y\) is called an ECP multifunction if for every \(\epsilon >0\), there exists \(\sigma >0\) such that each \(x \in X\) has a neighborhood \(N_x\) satisfying (1) \(d(f(z), f(x))<\sigma/2\) for every continuous function \(f: X \to Y\) with \(f(x) \in B_\epsilon (T(x))\) for each \(x \in X\) and \(z \in N_x\), and (2) \(d(y_1, y_2)<\epsilon\) for each \(y_1, y_2 \in \bigcap_{z \in N_x} B_\sigma (T(z))\). The authors prove the following theorem: Let \(X\) be a paracompact zero-dimensional space, \(Y\) a complete metric space and \(T: X \to 2^Y\) a multifunction. If there is an a.l.s.c.\ ECP multifunction \(S: X \to 2^Y\) satisfying (1) each \(S(x)\) is closed and \(S(x) \subset T(x)\) for each \(x \in X\), and (2) for each \(x \in X\), \(\overline{B}_\eta (S(x))\) is compact for some \(\eta>0\), then \(T\) admits a continuous selection.