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Let $X$ and $Y$ be disjoint normal topological spaces and, $A\subseteq X$ a closed subset with $f:A\rightarrow Y$ a continuous function. Please help me to show in detail that the pullback of $X$ and $Y$ which is the quotient space $X+_f Y$, is a normal space.

Many Thanks

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  • $\begingroup$ the pullback is the adjunction space? $\endgroup$ Apr 21, 2018 at 14:31
  • $\begingroup$ Sounds like you mean push out? $\endgroup$ Apr 21, 2018 at 21:28

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Sketch of proof, based on Dugundji (Topology, p. 146/147):

let $p$ be the quotient map from $X + Y$ to $X \cup_f Y$, the adjunction space of $X$ to $Y$ via $f$. If $F_1, F_2$ are disjoint and closed in $X \cup_f Y$, then $Y \cap p^{-1}[F_i]$ $i=1,2$ are closed and disjoint in $Y$, so there are $V_i$ open in $Y$ such that $Y \cap p^{-1}[F_i] \subseteq V_i, i=1,2$ and $\overline{V_1} \cap \overline{V_2} = \emptyset$. Then both $p[\overline{V_i}]$ are closed in $X \cup_f Y$ (check this!) and both $X\cap p^{-1}[F_i \cup p[\overline{V_i}]]$ are closed and disjoint in $X$, so these sets can be separated in $X$ by disjoint open subsets $U_i$ in $X$. Then check that for both $i$, $F_i \subseteq p[(U_i \setminus A) \cup V_i]$ and the latter sets are open and disjoint in $X \cup_f Y$.

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    $\begingroup$ An alternative proof is in G2 here $\endgroup$ Apr 21, 2018 at 18:06

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