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------------------------------------------------ Let f be a function from X to Y. If X is normal, which of the following conditions on f implies that Y is normal.

a. homeomorphism

b. continuous, open

c. continuous, onto

d. open, onto

e, open

Answer :

Final answer:

The condition that guarantees the normality of Y is a function that is both continuous and open. The correct option is (b).

Explanation:

To determine which condition on the function f implies that Y is normal, we need to analyze each option:

  1. Homeomorphism: A homeomorphism is a bijective function that is continuous and has a continuous inverse. While homeomorphisms preserve many topological properties, they do not guarantee the normality of the image set Y.
  2. Continuous, open: A function that is both continuous and open preserves the normality of the image set Y. This means that if f is continuous and open, and X is normal, then Y will also be normal.
  3. Continuous, onto: A function that is continuous and onto does not necessarily guarantee the normality of the image set Y.
  4. Open, onto: A function that is open and onto does not necessarily guarantee the normality of the image set Y.
  5. Open: A function that is open does not necessarily guarantee the normality of the image set Y.

Based on the analysis, the condition that guarantees the normality of Y is a function that is both continuous and open.

Learn more about conditions for a function to imply normality of a set here:

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