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Topology

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Comparing The Product And Box Topologies

TOPO-JGSYZN

Let $I$ be an index set and $\{X_i\}_{i\in I}$ an indexed family of topological spaces.

If $Y$ denotes $\Pi_{i\in I} X_i$ given the product topology and $Z$ denotes $\Pi_{i\in I} X_i$ given the box topology, which of the following are necessarily TRUE?

A

$Y=Z$ if $I$ is finite.

B

$Y\neq Z$ is $I$ is infinite.

C

Every open set of $Y$ is open in $Z$ as well.

D

The topology on $Z$ is finer than the topoology on $Y$.

E

The topology on $Z$ is coarser than the topology on $Y$.