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I read the statement that the Euler characteristic is always additive with respect to closed-closed union, which means that $\chi(X\sqcup Y) = \chi(X)+\chi(Y)$ if $X$ and $Y$ are closed.

And I read that this is not true with respect to closed-open union. Can someone give me a counterexample showing that this is not additive?

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    Are you sure that $\sqcup$ isn't a disjoint union? Otherwise, I don't believe the formula.2011-07-18

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E.g. $[0,1]=\{0\}\sqcup (0,1]$ and the Euler characteristics of each of the three spaces is $1$.

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    what is the reason for the property being true when $X$ and $Y$ are closed?2012-08-26