how to show boundary of either open or closed set is nowhere dense.

Question

how to show boundary of either open or closed set is nowhere dense.

i think we need to use baire category thm?

countable intersection of dense, closed set is once again a dense, closed (and nonempty)

Answer 1

Hint: use the expression $$ \partial A = \bar A - \text{int }A $$

Answer 2

Use mookid's hint. Suppose A is closed and its boundary contains an open $V$. Then $V$ is in A but not in the interior of A! Impossible, since???? (definition of the interior of a set...)