Topological classification of function spaces with the Fell topology. I. (Q471456)

Let \(X\) be a Tychonoff topological space, \(L\) a subset of the real line \(\mathbb R\) with the usual order and topology, and \(USC(X, L)\) (respectively, \(C(X, L)\)) the set of all upper semi-continuous maps (respectively, continuous maps) from \(X\) to the subspace \(L\) of \(\mathbb R\). For every \(f\in USC(X, L)\), the closed set \(\downarrow f = \{(x, t) \in X\times L : t \leq f(x)\}\subseteq X\times L\) is called the hypograph of \(f\). By \(\downarrow C_F(X)\) the authors denote the collection of the hypographs of all continuous maps from \(X\to [0, 1]\) with the Fell topology. In this interesting paper they prove the following main theorem: { Main Theorem.} For every weakly locally compact and hemicomapct infinite \(\aleph_0\)-space \(X\), there exists a space \(M^Q\) such that \(M^Q\approx Q\) and \(\downarrow C_F(X)\) can be imbedded to \(M^Q\) as a homotopy dense set.