Answer :
Final answer:
The condition on the function that implies Y is first countable is continuous, open.
Explanation:
In order to determine which conditions on the function imply that Y is first countable, we need to understand the concept of first countability and its preservation under certain conditions.
A topological space X is said to be first countable if for every point x in X, there exists a countable collection of open sets {U_n} such that for any open set U containing x, there exists an open set U_n in the collection such that x is contained in U_n and U_n is contained in U.
Now, let's analyze the given conditions:
- a. Homeomorphism: A homeomorphism is a bijective function between two topological spaces that preserves open sets. However, being a homeomorphism does not guarantee the preservation of first countability. Therefore, option a is not the correct answer.
- b. Open: If f is an open function, meaning it maps open sets to open sets, then it does not necessarily imply that Y is first countable. Therefore, option b is not the correct answer.
- c. Open, onto: If f is an open and onto function, then it does not necessarily imply that Y is first countable. Therefore, option c is not the correct answer.
- d. Continuous, onto: If f is a continuous and onto function, then it does not necessarily imply that Y is first countable. Therefore, option d is not the correct answer.
- e. Continuous, open: If f is a continuous and open function, then it preserves the first countability property. If X is first countable and f is continuous, then Y is also first countable. Therefore, option e is the correct answer.
Learn more about conditions for a function to imply first countability here:
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