Let X,Y1, Y2 be metric spaces. Let f: X--> Y1× Y2 be a map. Write f(x) = (f1(x), f2(x)). Show that f is continuous if and only if each of f1, f2 is continuous

Given the problem, we want to show that a map \( f: X \to Y_1 \times Y_2 \) is continuous if and only if each of \( f_1: X \to Y_1 \) and \( f_2: X \to Y_2 \) is continuous, where \( f(x) = (f_1(x), f_2(x)) \).

### Definitions:
- A map \( f: X \to Y_1 \times Y_2 \) is continuous if, for every open set \( U \subset Y_1 \times Y_2 \), \( f^{-1}(U) \subset X \) is open.
- The product space \( Y_1 \times Y_2 \) is equipped with the **product topology**, where the open sets are unions of sets of the form \( U_1 \times U_2 \), where \( U_1 \subset Y_1 \) and \( U_2 \subset Y_2 \) are open sets.
- \( f_1 \) and \( f_2 \) are continuous if the preimages of open sets in \( Y_1 \) and \( Y_2 \), respectively, are open in \( X \).

### Proof:

#### (⇐) If \( f_1 \) and \( f_2 \) are continuous, then \( f \) is continuous:

1. Let \( U \subset Y_1 \times Y_2 \) be an open set in the product topology. By definition of the product topology, we can assume \( U \) is of the form \( U_1 \times U_2 \), where \( U_1 \subset Y_1 \) and \( U_2 \subset Y_2 \) are open sets.
   
2. The preimage of \( U \) under \( f \) is given by:
   \[
   f^{-1}(U_1 \times U_2) = \{ x \in X : f(x) \in U_1 \times U_2 \} = \{ x \in X : f_1(x) \in U_1 \text{ and } f_2(x) \in U_2 \}.
   \]
   This can be written as:
   \[
   f^{-1}(U_1 \times U_2) = f_1^{-1}(U_1) \cap f_2^{-1}(U_2).
   \]
   
3. Since \( f_1 \) and \( f_2 \) are continuous by assumption, \( f_1^{-1}(U_1) \) and \( f_2^{-1}(U_2) \) are open sets in \( X \). Therefore, the intersection \( f_1^{-1}(U_1) \cap f_2^{-1}(U_2) \) is open in \( X \).

4. Since any open set in \( Y_1 \times Y_2 \) can be written as a union of sets of the form \( U_1 \times U_2 \), and the preimage of such sets is open, we conclude that \( f \) is continuous.

#### (⇒) If \( f \) is continuous, then \( f_1 \) and \( f_2 \) are continuous:

1. Consider any open set \( U_1 \subset Y_1 \). The set \( U_1 \times Y_2 \) is an open set in \( Y_1 \times Y_2 \) because it is the product of the open set \( U_1 \) in \( Y_1 \) and the entire space \( Y_2 \), which is open by definition.

2. Since \( f \) is continuous, the preimage of \( U_1 \times Y_2 \) under \( f \), which is:
   \[
   f^{-1}(U_1 \times Y_2) = \{ x \in X : f_1(x) \in U_1 \},
   \]
   is open in \( X \). This shows that \( f_1^{-1}(U_1) \) is open in \( X \), so \( f_1 \) is continuous.

3. Similarly, for any open set \( U_2 \subset Y_2 \), the set \( Y_1 \times U_2 \) is open in \( Y_1 \times Y_2 \). The preimage of \( Y_1 \times U_2 \) under \( f \), which is:
   \[
   f^{-1}(Y_1 \times U_2) = \{ x \in X : f_2(x) \in U_2 \},
   \]
   is open in \( X \). This shows that \( f_2^{-1}(U_2) \) is open in \( X \), so \( f_2 \) is continuous.

Thus, if \( f \) is continuous, then both \( f_1 \) and \( f_2 \) are continuous.

### Conclusion:
The function \( f: X \to Y_1 \times Y_2 \) is continuous if and only if both \( f_1: X \to Y_1 \) and \( f_2: X \to Y_2 \) are continuous.

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Would you like more details or have any further questions?

### Related Questions:
1. What is the product topology, and how is it defined?
2. Can we generalize this result to products of more than two metric spaces?
3. How does this result change if \( f_1 \) or \( f_2 \) are not continuous but still bounded?
4. Are there any counterexamples of discontinuous maps whose components are continuous?
5. How do we define continuity in topological spaces versus metric spaces?

### Tip:
Continuity in product spaces often breaks down into examining the behavior of individual components, which is a powerful tool in both analysis and topology.

Let X, Y1, Y2 be metric spaces. Let f: X --> Y1 × Y2 be a map. Write f(x) = (f1(x), f2(x)). Show that f is continuous if and only if each of f1, f2 is continuous.

Explore the detailed proof showing that a map f: X --> Y1 × Y2 is continuous if and only if its components f1: X --> Y1 and f2: X --> Y2 are continuous. This problem involves metric spaces, product topology, and the continuity of open sets. Suitable for advanced undergraduate students.

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