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Let $X$ denote the product of countably many copies of [$0,1$] . we let $X_1$ denote the set $X$ equipped with the box topology and let$X_2$ denote the se $X$ equipped with the product topology. Then
(1) $X_1$ is compact and separable.
(2) $X_2$ is compact and separable.
(3) $X_1$ and $X_2$ are both compact
(4) Neither $X_1$ nor $X_2$ is separable

Box topology is not compact nor separable but product topology is both. [0,1] is compact metric space so separable. So 1 is correct.but not sure about the others.can anybody help me .thanks.

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The first sentence of your second paragraph is enough to determine that only (2) is true. (In fact, you only need to see that (2) is true and that $X_1$ is not compact.) –  David Mitra Dec 19 '12 at 16:31
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1 Answer

up vote 1 down vote accepted

Compactness of $X_2$ follows from Tychonoff's theorem.

The Box topology is not compact because the following is an open cover with no finite subcover: $\{ \prod_{i \in I} (0+\frac1n, 1-\frac1n)\}_{n \in \mathbb N}$.

Hence (3) and (1) are false.

The product topology is separable hence (4) is false and (2) is true.

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