TITLE: How do you prove that $\mathsf{bd}(\mathsf{bd}(\mathsf{bd}(W)))= \mathsf{bd}(\mathsf{bd}(W))$
QUESTION [6 upvotes]: $\newcommand{\bd}{\operatorname{bd}}$Prove that the $\bd(\bd(\bd(W)))=\bd(\bd(W))$ where $W$ is a subset of the topological space $(X,\mathscr{T})$.

REPLY [3 votes]: $\DeclareMathOperator{\bd}{bd}$Prove that:
if $W$ is any subset of $X$, then $\bd(W)$ is closed;
if $Y$ is any closed subset of $X$, then $\bd(\bd(Y)) = \bd(Y)$.