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In the book Sheaf Theory, by Bredon (edition from 1997), Theorem 14.1, he writes a natural exact sequence, which, in some nice cases, leads to the Künneth formula. Do we have any reason to believe that the maps in the sequence are continuous?

I'm in a situation that I can compute the cohomology spaces associated to a differential complex and, after using the Künneth formula, I obtain that the spaces are Hausdorff spaces. Unfortunately, I don't know if I can deduce that the cohomology spaces are Hausdorff without knowing if the Künneth isomorphism are continuous.

I would greatly appreciate any suggestion of literature that deals with continuity of the isomorphism obtained Künneth formula.

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    $\begingroup$ Do you mean in Chapter V? If so, I'm not exactly sure what you mean by cohomology spaces and continuity. I'm seeing a short exact sequence of (products of) homology groups. I'm not sure what space maps you're considering. Would you please clarify. $\endgroup$
    – Greg Friedman
    Commented Apr 16, 2023 at 5:48

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