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Answer :
Final answer:
If XxY is compact, then both X and Y are compact.
Explanation:
To determine whether X and Y are compact when XxY is compact, we need to consider the projection function. The projection function maps each point in the Cartesian product to its corresponding factor. In this case, the projection function is both continuous and onto.
By definition, a function is continuous if the pre-image of every open set is open. Since the projection function is continuous, it preserves the compactness of the Cartesian product. Therefore, if XxY is compact, both X and Y must be compact.
Furthermore, the projection function is onto, meaning that every point in the Cartesian product has a corresponding point in each factor. This property does not directly affect the compactness of X and Y, but it is a characteristic of the projection function.
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