If $ X \times Y $ is separable, then $ X $ and $ Y $ are separable. This is because the projection function is:

A. onto
B. continuous and onto
C. continuous
D. open and onto
E. open

To determine if X and Y are separable, we need to consider the properties of the projection function. The projection function takes an element from the Cartesian product XxY and returns the corresponding element from either X or Y. In order for X and Y to be separable, the projection function must be continuous and onto.

A continuous function is one that preserves the topological structure of the sets. It means that small changes in the input result in small changes in the output. In this case, the projection function should not introduce any abrupt changes or discontinuities.

The projection function also needs to be onto, which means that every element in X and Y should have a corresponding element in XxY. This ensures that no elements are left out or lost during the projection process.

Therefore, the correct answer is c. continuous.

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If \( X \times Y \) is separable, then \( X \) and \( Y \) are separable. This is because the projection function is:A. onto B. continuous and onto C. continuous D. open and onto E. open

Answer :

Final answer:

Explanation:

Other Questions

If \( X \times Y \) is separable, then \( X \) and \( Y \) are separable. This is because the projection function is:

A. onto
B. continuous and onto
C. continuous
D. open and onto
E. open