For a metrizable space X and a finite measure space (Ω, $𝔐$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $𝔐$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by $USC{C}_B\left(X\right)$ the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify $\phi \in USC{C}_B\left(X\right)$ with its graph which is a closed subset of X × ℝ. The space $USC{C}_B\left(X\right)$ admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then $USC{C}_B\left(X\right)$ is homeomorphic to a...

For A ⊂ I = [0,1], let $L_A$ be the set of continuous real-valued functions on I which vanish on a neighborhood of A. We prove that if A is an analytic subset which is not an $F_{\sigma }$ and whose closure has an empty interior, then $L_A$ is homeomorphic to the space of differentiable functions from I into ℝ.