theory (as presented, e.g. in [@DGS; @KH; @Walters]). For every result which already exists in the literature the statement itself includes the reference to the original source. But some results we provide are restatements or compilations for which no single reference is appropriate. In these cases we only include the author’s name (if such an attribution is possible) in the statement and cite the relevant origins in the preceding paragraph. To make our presentation complete we also introduce a few original results. They mostly come from the second named author’s Master’s Thesis written at the Jagiellonian University in Kraków under supervision of the first named author. In particular, the results in Section \[sec:app\] (on connections between the almost product property and shadowing) namely Theorems \[thm:shd:aps\] and \[thm:40\], Corollaries \[cor:shad\] and \[cor:41\] and most Examples have not been published before.

Acknowledgements {#acknowledgements .unnumbered}
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We would like to thank the referee for his/her careful reading and constructive comments. We are also grateful to Vaughn Climenhaga and Ronnie Pavlov for many discussions on the subject of the paper.

Dominik Kwietniak was supported by the Polish National Science Centre (NCN) grant no. 2013/08/A/ST1/00275; the research of Martha [Ł]{}cka was supported by the Polish 