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Answer :
Final answer:
If XxY is separable, then X and Y are separable because the projection function is open and onto. Hence the correct answer is option c.
Explanation:
To determine the properties of X and Y when XxY is separable, we need to consider the properties of the projection function. The projection function takes an element from the Cartesian product XxY and returns its first or second component. In this case, we are interested in whether the projection function is continuous, onto, open, or a combination of these properties.
The fact states that the projection function is open and onto. This means that the projection function maps XxY onto X and Y, and it also preserves open sets. In other words, if XxY is separable, then X and Y are also separable because the projection function is open and onto.
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