- Thread starter
- #1
- Mar 10, 2012
- 835
Suppose $X=A_1 \cup A_2 \cup \ldots ,$ where $A_n \subseteq \text{ Interior of } A_{n+1}$ for each $n$. If $f:X \rightarrow Y$ is a function such that $f|A_n:A_n \rightarrow Y$ is continuous with respect to the induced topology on $A_n$, show that $f$ itself is continuous.
Now here's what I think can be useful in solving the problem.
Let $V$ be open in $Y$. Then $f^{-1}(V)=\bigcup_{i=1}^{i=\infty}{f|A_i}^{-1}(V)$, where each inverse image in the RHS is open in its corresponding restricted space.
I can't see how to go further. Please help.
Now here's what I think can be useful in solving the problem.
Let $V$ be open in $Y$. Then $f^{-1}(V)=\bigcup_{i=1}^{i=\infty}{f|A_i}^{-1}(V)$, where each inverse image in the RHS is open in its corresponding restricted space.
I can't see how to go further. Please help.