Question:
Consider the following statements:
P: Every compact metrizable topological space is separable.
Q: Every Hausdorff topology on a finite set is metrizable.
Then,
Consider the following statements:
P: Every compact metrizable topological space is separable.
Q: Every Hausdorff topology on a finite set is metrizable.
Then,
P: Every compact metrizable topological space is separable.
Q: Every Hausdorff topology on a finite set is metrizable.
Then,
Show Hint
In topology, compact metrizable spaces are separable, and finite Hausdorff spaces are always metrizable.
Updated On: Jan 7, 2026
- both P and Q are TRUE
- P is TRUE and Q is FALSE
- P is FALSE and Q is TRUE
- both P and Q are FALSE
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The Correct Option is A
Solution and Explanation
- P: Every compact metrizable topological space is separable. This statement is TRUE. A metrizable space is one that has a topology induced by a metric, and every compact metrizable space is separable. In fact, separability is a key property of compact metrizable spaces.
- Q: Every Hausdorff topology on a finite set is metrizable. This statement is also TRUE. Any finite Hausdorff space is discrete, and any discrete space is metrizable with the discrete metric. Therefore, the space is metrizable.
Thus, both P and Q are true, making the correct answer (A).
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