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:
- 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.
- 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.
- Continuous, onto: A function that is continuous and onto does not necessarily guarantee the normality of the image set Y.
- Open, onto: A function that is open and onto does not necessarily guarantee the normality of the image set Y.
- 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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