High School

Let \( f \) be a function from \( X \) to \( Y \). If \( X \) is first countable, which of the following conditions on \( f \) implies that \( Y \) is first countable?

A. open, onto
B. continuous, onto
C. open
D. continuous, open
E. homeomorphism

Answer :

Final answer:

The conditions that imply Y is first countable are: continuous and onto, and homeomorphism.

Explanation:

To determine which conditions on the function f imply that Y is first countable, we need to understand the properties of functions that preserve the first countability of a space.

First countability is a property of topological spaces, which means it is related to the open sets in the space. A topological space is said to be first countable if every point in the space has a countable neighborhood basis. A neighborhood basis for a point x is a collection of open sets such that every open set containing x contains at least one set from the collection.

Now, let's consider the given options:

  1. a. open, onto: This condition does not guarantee that Y is first countable. Being open and onto does not necessarily preserve the first countability of a space.
  2. b. continuous, onto: This condition implies that Y is first countable. A continuous function preserves the first countability of a space. Therefore, if f is continuous and onto, then Y is first countable.
  3. c. open: This condition does not guarantee that Y is first countable. Being open does not necessarily preserve the first countability of a space.
  4. d. continuous, open: This condition does not guarantee that Y is first countable. Being continuous and open does not necessarily preserve the first countability of a space.
  5. e. homeomorphism: This condition implies that Y is first countable. A homeomorphism is a bijective function that is continuous and has a continuous inverse. Homeomorphisms preserve the topological properties of spaces, including first countability.

Therefore, the conditions that imply Y is first countable are:

  • Continuous and onto (Option b)
  • Homeomorphism (Option e)

Learn more about conditions for a function to imply first countability here:

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