Continuity of interpolations (Q864063)

All topological spaces are assumed to be Tychonoff. As usual, \(\mathbb{R}\) stands for the real line equipped with the Euclidian topology. Given a topological space \(X\), \(F(X\times \mathbb{R})\) stands for the hyperspace of all finite subsets of the product space \(X\times \mathbb{R}\) endowed with the Vietoris topology. Let \(S(X)\) be the subspace of \(F(X\times \mathbb{R})\) defined by \[ S(X)=\{\{(x_{1},r_{1}),\ldots , (x_{n},r_{n})\} : x_{i}\neq x_{j} \text{ for } i\neq j\}\, . \] For each \(n=1,2,\dots \), \(F_{n}(X\times \mathbb{B})\) means the set of the elements of \(F(X\times \mathbb{R})\) which have at most \(n\) points and \(S_{n}(X)=S(X)\cap F_{n}(X\times \mathbb{R})\). For a point \(D=\{(x_{1},r_{1}),\ldots , (x_{n},r_{n})\}\in S(X)\), a function \(f_{D}\) in the usual Banach space \(C(X)\) of all real-valued continuous functions on \(X\) with the supremum norm is called an interpolation function for \(D\) if \[ f_{D}(x_{1})=r_{1}, f_{D}(x_{2})=r_{2},\ldots , f_{D}(x_{n})=r_{n}. \] If in addition the map \(g:S(X)\rightarrow C(X)\) defined by \(g(D)=f_{D}\) is continuous, then \(g\) is said to be a continuous interpolation of \(X\). In the case \(g\) satisfies the weaker condition that the restriction \(g| _{S_{n}(X)\setminus S_{n-1}(X)}\) is continuous for each \(n=1,2,\dots \), we say that \(g\) is weakly continuous. The author studies the existence of (weakly) continuous interpolations for several classes of spaces. The author proves the following facts: (a) Every metrizable space has a weakly continuous interpolation, (b) the ordered space \(\omega_{1}\) of the first uncountable ordinal and the one-point compactification of the discrete space \(D(\omega_{1})\) don't have a weakly continuous interpolation, (c) the one-point Lindelöfication of the discrete space \(D(\omega_{1})\) has a continuous interpolation, (d) there exist almost disjoint families \(\mathcal{A}\) and \(\mathcal{B}\) of cardinality \(2^{\omega}\) of subsets of \(\omega\) such that the Isbell-Mrówka space \(\Psi(\mathcal{A})\) has a weakly continuous interpolation but the Mrówka space \(\Psi (\mathcal{B})\) does not. The author also shows that ``have a weakly continuous interpolation'' is not a finite productive property and also obtains some relationship between this property and some cardinal invariants: (a) If \(X\) is a countably compact space which has a weakly continuous interpolation, then the tightness of \(X\) is countable, (b) let \(X\) be a space such that \(X\times (\omega +1)\) has a weakly continuous interpolation. Then the pseudocharacter of \(X\) is countable, and (c) if an infinite space \(X\) has a weakly continuous interpolation, then the density of \(X\) is larger or equal to the minimum cardinality of separating families of \(X\) (a family \(\mathcal{F}\subset C(X)\) is called separating if for any distinct points \(x\) and \(y\) in \(X\) there exists \(f\in \mathcal{F}\) such that \(f(x)\neq f(y)\)). As a consequence of these results the uncountable product space \(\{0,1\}^{\omega_{1}}\) does not have a weakly continuous interpolation and the Stone-Čech compactification \(\beta \omega\) of the countable infinite discrete space \(\omega\) does not have a continuous interpolation.