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Jobs.Q:
Using different notations for topologies on product spaces
Suppose that a set $X$ has two different topologies. Does the product topology on the product space $X^2$ (whose basic open sets are the product sets $Z = \prod_{i \in I} Z_i$ where $Z_i = \{(x_j)_{j \in I}: x_i \in Z_i\}$) inherit one of these topologies from $X$?
If not, what about the “inductive limit topology” on $X^\mathbb{N}$? (whose basic open sets are the sets $Z = \prod_{i \in \mathbb{N}} Z_i$ where $Z_i = \{(x_j)_{j \in \mathbb{N}}: x_i \in Z_i\}$)
A:
There is no “first” topology. The product topology is built by first taking the disjoint union topology on $X\times X$, then taking the product topology on each coordinate. So you have the disjoint union topology on the disjoint union, then the product topology on the product.
But if you want a space where the subsets are determined by the topology induced on the factors, then the product space is not going to be one of them. Suppose that $X$ has a basis $\mathcal{B}_0$, and the topology is the subspace topology. Let $\mathcal{B}$ be any basis of the product space. Then for every $B\in\mathcal{B}$, there is an open neighborhood $U_B$ of $B$ in $X
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