Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.