I was reading the Munkres and found a theorem (namely, the Theorem 19.3) and I didn't get it:
Theorem 19.3 Let $A_{\alpha}$ be a subspace of $X_{\alpha}$ , for each $\alpha \in J$. Then $\prod A_{\alpha}$ is a subspace of $\prod X_{\alpha}$ if both products are given the box topology, or if both products are given the product topology.
How one shows that a given set is a subspace of other one? I think this is what I don't get in this theorem.
Thanks in advance.
Best Answer
You must show that $U$ is an open subset of $\prod A_\alpha,$ if and only if there is some open subset $V$ of $\prod X_\alpha$ such that $U=V\cap\prod A_\alpha$ (meaning $U$ is open in the subspace topology on $\prod A_\alpha$ induced by $\prod X_\alpha$). Do you know the definitions of the box and product topologies?