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Topology

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Products of Spaces with the Discrete Topology

TOPO-DIVKQB

For each $n\in\mathbb{N}$, let $X_n$ denote the space of integers $\mathbb{Z}$ endowed with the discrete topology.

Which of the following are basic open sets in $\Pi_{n=1}^\infty X_n$ given the product topology?

Select ALL that apply.

A

$\Pi_{n=1}^\infty A_n$, where for each $n\in\mathbb{N}$, $A_n = \{-1,2\}$.

B

$\Pi_{n=1}^\infty A_n$, where for each $n\in\mathbb{N}$, $A_n = \{-1,2\}$ if $n=1$ and $A_n=X$ if $n\neq 1$.

C

$\Pi_{n=1}^\infty A_n$, where for each $n\in\mathbb{N}$, $A_n = \{-1,2\}$ if $n\neq 1$ and $A_n=X$ if $n=1$.

D

$\Pi_{n=1}^\infty A_n$, where for each $n\in\mathbb{N}$, $A_n=X$.

E

$\Pi_{n=1}^\infty A_n$, where for each $n\in\mathbb{N}$, $A_n = \{-1,2\}$ if $n$ is negative and $A_n=X$ if $n$ is non-negative.