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An exercise asks me to do the following:

$(x_1,\tau_1), (X_2,\tau_2)$ are two topological spaces. $A_i\in X_i, i=1,2.$ Show that $\partial(A_1\times A_2)=(\partial A_1\times A_2)\cup(A_1\times \partial A_2)$, given that $X_1\times X_2$ has the product topology.

I think the result is wrong. I believe it should be $\partial(A_1\times A_2)=(\partial A_1\times \bar{A_2})\cup(\bar{A_1}\times \partial A_2)$, where bar denotes closure and $\partial$ denotes boundary.

Am I right?

Ma Joad
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    Just as an example, take $X_i = \Bbb R$ and $A_i = (0,1)$. Then $\partial (A_1\times A_2)$ should be the boundary of the unit square, yet with the given definition the corners aren't included. – Arthur May 21 '19 at 11:28
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    The answer to the linked question gave a proof of $\partial(A_1\times A_2)=(\partial A_1\times \bar{A_2})\cup(\bar{A_1}\times \partial A_2)$ (so this should be the correct product rule) – YuiTo Cheng May 21 '19 at 12:08

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