Abstract. The paper deals with Ascoli spaces Cp(X) and C k (X) over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X for which any compact subset K of C k (X) is evenly continuous, essentially includes the class of k R -spaces. First we prove that if Cp(X) is Ascoli, then it is κ-Fréchet-Urysohn. If X is cosmic, then Cp(X) is Ascoli iff it is κ-Fréchet-Urysohn. This leads to the following extension of a result of Morishita: If for aČech-complete space X the space Cp(X) is Ascoli, then X is scattered. If X is scattered and stratifiable, then Cp(X) is an Ascoli space. Consequently: (a) If X is a complete metrizable space, then Cp(X) is Ascoli iff X is scattered. (b) If X is aČech-complete Lindelöf space, then Cp(X) is Ascoli iff X is scattered iff Cp(X) is Fréchet-Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent:is an Ascoli space. The Asoli spaces C k (X, I) are also studied.
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In this paper we work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of Markov operators on a Hilbert space and it is characterized in terms of iterated degree distributions, homomorphism density of trees, weak isomorphism of a conditional expectation with respect to invariant sub-σ-algebras and isomorphism of certain quotients of given graphons. Our proofs use a weak version of the mean ergodic theorem, and correspondences between objects such as Markov projections, sub-σ-algebras, conditional expectation, etc. That also provides an alternative proof for the characterizations of fractional isomorphism of graphs without the use of Birkhoff-von Neumann Theorem.
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