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单词 BoundaryOfAClosedSetIsNowhereDense
释义

boundary of a closed set is nowhere dense


Let A be closed. In general, the boundary of a set is closed. So it suffices to show that A has empty interior.

Let UA be open. Since AA¯=A, this implies that UA. Since int(A) is the largest open subset of A, we must have Uint(A). Therefore UAint(A). But Aint(A)=(A¯-int(A))int(A)=, so U=.

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更新时间:2025/5/4 5:41:03