I know that boundary of union is subset of union of boundaries. When does the equality hold? Can you give examples where there is equality and where there fails to be equality?
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Equality certainly holds when the subsets are either equal or open and disjoint. For an example of equality failing, think about two partially intersecting disks. – BHT Mar 22 '19 at 02:02
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@EthanMacBrough, does it have to be open? I'm thinking of a similar example to two partially intersecting disks, if we take two disjoint closed disks, then the equality still holds – dxdydz Mar 22 '19 at 02:10
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@dxdydz Consider $[0,1]\times [0,1]$ and $(1,2]\times[0,1]$. Then the boundary of the union is a single rectangle whereas the union of boundaries is two adjoined squares. – BHT Mar 22 '19 at 02:30
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I should clarify; the result may still hold even if the sets are not open, as in the question @Mariah linked. But in general being disjoint alone is insufficient. – BHT Mar 22 '19 at 02:32