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Apr 19, 2022 at 12:31 comment added user14334 After some thought I am still unsure why the long exact sequence should hold for all $X$ locally compact and Hausdorff. Looking at this post which asks the same question, one commenter argues that this should be true if $X$ is what they call HLC- and I am more or less persuaded by them. Another argues the sequence should be exact for general locally compact Hausdorff $X$ by an excision argument by which I am not fully convinced.
Mar 23, 2022 at 20:40 history answered Dan Petersen CC BY-SA 4.0