I am writing a paper on K-analytic spaces and need the following known characterization.



> 
> **Theorem.** For a regular topological space $X$ the following conditions are equivalent:
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> 
> (1) $X$ is a continuous image of a Lindelof Cech-complete space;
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> (2) $X$ is the image of a Polish space under an upper semicontinuous compact-valued map.
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> 


The condition (2) is usually taken as the definition of K-analytic spaces.


On the other hand, the equivalent condition (1) is shorter and intuitively is a better generalization of the analytic spaces (which are defiend as continuous images of Polish spaces).


The only book in which (1) is taken as the definition of K-analycity is the book ["General Topology, III"](https://books.google.com.ua/books?id=DhvpCAAAQBAJ&pg=PA37&lpg=PA37&dq=K-analytic%20spaces%20continuous%20images%20of%20Lindelof%20Cech-complete%20spaces&source=bl&ots=nZTgeQyMtG&sig=ACfU3U2_Y9anAvuI-imSJdNqiNPPLD_hTA&hl=uk&sa=X&ved=2ahUKEwjP4Mqh-9nhAhUkl4sKHdL5BIYQ6AEwCnoECAgQAQ#v=onepage&q=K-analytic&f=false) of Arhangelski (see page 37). But this book does not prove that these two conditions are indeed equivalent.


So I asking the MO-community for help with finding a proper reference to the above theorem (in order to avoid writing a proof which is rather standard).



A reference for a (folklore?) characterization of K-analytic spaces
See "Descriptive Topology" by R.Hansell in "Recent Progress in General Topology" (1992), p. 281-282


