Let $\{{A_n}\}$ be the closed subsets of $X$, such that ${A_n} \subset \operatorname{Int}{A_{n + 1}}$ and $ \cup {A_n} = X$, if $A_1$ and all $\operatorname{cl}(A_n-A_{n-1})$ have the covering dimension at most $m$, does $X$ have the covering dimension at most $m$?
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{\rm Int} Arather than\rm{Int} A, since\rmdoesn't take a parameter — it is simply a switch that changes the current font. – Davide Cervone Apr 20 '12 at 12:15