Exercise 4, § 3

Let $Y, Z$ be two subspaces of a topological space $X$ such that $X = Y \cup Z$ and such that $\overline{A} \cap B = \overline{B} \cap A = \varnothing$, where $A = Y \cap \complement Z$ and $B = Z \cap \complement Y$.

a) Show that if $M$ is any subset of $X$ then the closure of $M$ in $X$ is the union of the closure of $M \cap Y$ in $Y$ and the closure of $M \cap Z$ in $Z$. Deduce that if $M \cap Y$ is closed (resp. open) in $Y$ and $M \cap Z$ is closed (resp. open) in $Z$, then $M$ is closed (resp. open) in $X$.

b) Let $f$ be a mapping of $X$ into a topological space $X'$. Show that if $f|Y$ and $f|Z$ are continuous then so is $f$.