The following is an example problem from the above-mentioned text:
"Let $J^+$ = [n, n is a positive integer] and let $I_n = [0, 1/n]$. Then $P = (\pi_{n \; \in \; J^+}, I_n)$ is a topological space in the product topology. We introduce a metric on the point set of $P$ as follows..."
My question is the following:
What product space corresponds to the product topology mentioned above?
EDIT: The notation $(\pi_{n \; \in \; J^+}, I_n) $ seems to suggest that said product topology corresponds to the set $I_n$. If this is the case, how can $I_n$ be treated as a product space?
Thanks in advance!
It is possible (but notationally very weird) that what is being provided is a collection of canonical projections and their image spaces. Starting from that guess, your product space is $J^+$-indexed sequences, where the $n^\text{th}$ element of the sequence is an element of $I_n$.
A slight reduction in notational weirdness would be $$( (\pi_n, I_n))_{n \in J^+} \text{,} $$ which more clearly indicates a sequence indexed by $J^+$together with canonical projection - images space pairs. In either case, this mode of specification and the one in the Question are very category theoretic in that the specification provides the data necessary to construct the commutative diagram for the universal property of a product space.
As others have written in comments, "$ P = \prod_{n \in J^+} I_n $ with element-wise canonical projections" would be a far less weird way to specify this space.