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A space Y is called a free space if for each compactum X the set of maps with hereditarily indecomposable fibers is a dense $G_{\delta }$-subset of C(X,Y), the space of all continuous functions of X to Y. Levin proved that the interval I and the real line ℝ are free. Krasinkiewicz independently proved that each n-dimensional manifold M (n ≥ 1) is free and the product of any space with a free space is free. He also raised a number of questions about the extent of the class of free spaces. In this paper we will...

The space $\mathscr{H}$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold $X$ is an infinite dimensional symmetric space whose geodesics ${\omega }_t$ are solutions of a homogeneous complex Monge-Ampère equation in $A\times X$, where $A\subset ℂ$ is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials $\varphi (t,z)$ of ${\omega }_t$ may be approximated in a weak $C^0$ sense by geodesics ${\varphi }_N(t,z)$ of the finite dimensional symmetric space of Bergman metrics of height $N$. In this article we prove that ${\varphi }_N(t,z)\to \varphi (t,z)$ in $C^2([0,1]\times X)$ in the case of...