### A categorical proof of the equivalence of local compactness and exponentiability in locale theory

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Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.

We prove that for each countably infinite, regular space X such that ${C}_{p}\left(X\right)$ is a ${Z}_{\sigma}$-space, the topology of ${C}_{p}\left(X\right)$ is determined by the class ${F}_{0}\left({C}_{p}\left(X\right)\right)$ of spaces embeddable onto closed subsets of ${C}_{p}\left(X\right)$. We show that ${C}_{p}\left(X\right)$, whenever Borel, is of an exact multiplicative class; it is homeomorphic to the absorbing set ${\Omega}_{\alpha}$ for the multiplicative Borel class ${M}_{\alpha}$ if ${F}_{0}\left({C}_{p}\left(X\right)\right)={M}_{\alpha}$. For each ordinal α ≥ 2, we provide an example ${X}_{\alpha}$ such that ${C}_{p}\left({X}_{\alpha}\right)$ is homeomorphic to ${\Omega}_{\alpha}$.

We construct two examples of infinite spaces X such that there is no continuous linear surjection from the space of continuous functions ${c}_{p}\left(X\right)$ onto ${c}_{p}\left(X\right)$ × ℝ$.Inparticular,$cp(X)$isnotlinearlyhomeomorphicto$cp(X)$\times \mathbb{R}$. One of these examples is compact. This answers some questions of Arkhangel’skiĭ.

A metric space $\langle X,d\rangle $ is called a $UC$ space provided each continuous function on $X$ into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that $UC$ spaces play relative to the compact metric spaces.

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $$\overline{C}$$ (X) of C(X) such that the pair ($$\overline{C}$$ (X), C(X)) is homeomorphic to (Q, s). In case...

A space $X$ is functionally countable if $f\left(X\right)$ is countable for any continuous function $f:X\to \mathbb{R}$. We will call a space $X$ exponentially separable if for any countable family $\mathcal{F}$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap \mathcal{G}\ne \varnothing $ whenever $\mathcal{G}\subset \mathcal{F}$ and $\bigcap \mathcal{G}\ne \varnothing $. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has...

We prove a non-archimedean Dugundji extension theorem for the spaces ${C}^{*}(X,\mathbb{K})$ of continuous bounded functions on an ultranormal space $X$ with values in a non-archimedean non-trivially valued complete field $\mathbb{K}$. Assuming that $\mathbb{K}$ is discretely valued and $Y$ is a closed subspace of $X$ we show that there exists an isometric linear extender $T:{C}^{*}(Y,\mathbb{K})\to {C}^{*}(X,\mathbb{K})$ if $X$ is collectionwise normal or $Y$ is Lindelöf or $\mathbb{K}$ is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace $Y$ of an ultraregular...