Characteristic property of the disjoint union topology/Proof

Suppose $f:\coprod_{\alpha\in I}X_\alpha\rightarrow Y$ is continuous $\implies$ $\forall\beta\in I\big[f\big\vert_{X_\beta^*}:X_\beta^*\rightarrow Y$ is continuous$\big]$

Let $\beta\in I$ be given.

• Let $U\in\mathcal{K}$ be given (so $U$ is an open set in $Y$)
  • By hypothesis, $f^{-1}(U)\in\mathcal{J}$ (where $\mathcal{J}$ is the topology on $\coprod_{\alpha\in I}X_\alpha$)
  • This means $\forall\gamma\in I\exists V\in\mathcal{J}_\gamma[f^{-1}(U)\cap i_\gamma(X_\gamma)=V]$
  • Thus $\exists V\in\mathcal{J}_\beta[f^{-1}(U)\cap i_\beta(X_\beta)=V]$
  • But:
    • $f\big\vert_{X_\beta^*}^{-1}(U)=f^{-1}(U)\cap X_\beta^*$ Caution:This really ought to have its own proof
  • So for $V$ being the image of some open set under the canonical injection, $i_\beta$

Work required:

1. Some sort of homeomorphism between $(X_\alpha,\mathcal{J}_\alpha)$ and the subspace $X_\alpha^*:=i_\alpha(X_\alpha)$ is needed