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5 hours ago comment added Martin Brandenburg @JamesEHanson Yes Todd has proven there that it is cocomplete. It should be noted that both Todd and the authors of the paper actually mean continuous open maps. It seems that some authors assume this, but I find this is a mistake, it causes unnecessary confusion. See also Wikipedia.
6 hours ago comment added James E Hanson As I discussed in an answer here, Bezhanishvili and Kornell showed that in the category $\mathrm{Top}_{\mathrm{open}}$ of topological spaces with open maps, the product of Sierpiński space with itself fails to exist. Is this category cocomplete?
7 hours ago history asked Martin Brandenburg CC BY-SA 4.0