Question

Let $f:A\rightarrow \Pi _{\alpha \epsilon J} X_{\alpha}$ be given by the equation $f(a)=(f_{\alpha} (a))_{\alpha \epsilon J}$ , where $f_{\alpha} :A \rightarrow X_{\alpha}$ for each$\alpha$. Let $\Pi_{\alpha \epsilon J} X_{\alpha}$ have the product topology. Then show that the function f is continuous if and only if each function $f_{\alpha}$ is continuous.

Answer 1

Let $f:A\rightarrow \Pi _{\alpha \epsilon J} X_{\alpha}$ be given by the equation $f(a)=(f_{\alpha} (a))_{\alpha \epsilon J}$ , where $f_{\alpha} :A \rightarrow X_{\alpha}$ for each$\alpha$. Let $\Pi_{\alpha \epsilon J} X_{\alpha}$ have the product topology. Then show that the function f is continuous if and only if each function $f_{\alpha}$ is continuous.