An old question of Arhangel’skii asks if the Menger property of a Tychonoff space X is preserved by homeomorphisms of the space $C_p(X)$ of continuous real-valued functions on X endowed with the pointwise topology. We provide affirmative answer in the case of linear homeomorphisms. To this end, we develop a method of studying invariants of linear homeomorphisms of function spaces $C_p(X)$ by looking at the way X is positioned in its (Čech–Stone) compactification.

We study spaces of continuous functions and sections with domain a paracompact Hausdorff k-space $X$ and range a nilpotent CW complex $Y$, with emphasis on localization at a set of primes. For $\mathop {\rm map}\nolimits _\phi (X,\,Y)$, the space of maps with prescribed restriction $\phi$ on a suitable subspace $A\subset X$, we construct a natural spectral sequence of groups that converges to $\pi _*(\mathop {\rm map}\nolimits _\phi (X,\,Y))$ and allows for detection of localization on the level of $E^2$. Our applications extend and unify the previously known results.

In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y, it is proved that the following three conditions are equivalent: (1) the Scott space $\Sigma \mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y, the function space $\mathbb{C}(X, Y)$ of all continuous mappings from X to Y equipped with the Isbell topology is H-sober; (3) for every H-sober space Y, the Isbell topology on $\mathbb{C}(X, Y)$ has property S with respect to H. One immediate corollary is that for a $T_{0}$ space X, Y is a d-space (resp., well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is a d-space (resp., well-filtered space). It is shown that for any $T_0$ space X for which the Scott space $\Sigma \mathcal O(X)$ is non-sober, the function space $\mathbb{C}(X, \Sigma 2)$ equipped with the Isbell topology is not sober. The function spaces $\mathbb{C}(X, Y)$ equipped with the Scott topology, the compact-open topology and the pointwise convergence topology are also discussed. Our study also leads to a number of questions, whose answers will deepen our understanding of the function spaces related to H-sober spaces.

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $K_{i}$ be a locally compact (Hausdorff) topological space and let ${\mathcal{H}}_{i}$ be a closed subspace of ${\mathcal{C}}_{0}(K_{i},\mathbb{F})$ such that each point of the Choquet boundary $\operatorname{Ch}_{{\mathcal{H}}_{i}}K_{i}$ of ${\mathcal{H}}_{i}$ is a weak peak point. We show that if there exists an isomorphism $T:{\mathcal{H}}_{1}\rightarrow {\mathcal{H}}_{2}$ with $\left\Vert T\right\Vert \cdot \left\Vert T^{-1}\right\Vert <2$, then $\operatorname{Ch}_{{\mathcal{H}}_{1}}K_{1}$ is homeomorphic to $\operatorname{Ch}_{{\mathcal{H}}_{2}}K_{2}$. We then provide a one-sided version of this result. Finally we prove that under the assumption on weak peak points the Choquet boundaries have the same cardinality provided ${\mathcal{H}}_{1}$ is isomorphic to ${\mathcal{H}}_{2}$.

An asymptotically orthonormal sequence is a sequence that is nearly orthonormal in the sense that it satisfies the Parseval equality up to two constants close to one. In this paper, we explore such sequences formed by normalized reproducing kernels for model spaces and de Branges–Rovnyak spaces.

We prove, among other things, that if ${{C}_{p}}\left( X \right)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces.

Let X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?