First Course in Topology: Countable sets.

Question

Consider nonempty sets $X$ and $Y$, and a function $f:X\rightarrow Y$. Suppose the inverse image of $Y$ under $f$ is countable for each element of $Y$ and assume $Y$ is countable. Prove that $X$ is countable.

Did you mean that "the inverse image of each element of $Y$ is countable"? It's unclear as currently written. — vociferous_rutabaga, Aug 17 '14 at 12:19
Also, Welcome to MSE! When you post a question, please try to include some context and your attempt at a solution. See here for general guidelines. — vociferous_rutabaga, Aug 17 '14 at 12:22

score 3 · Answer 1 · edited Apr 13 '17 at 12:20

3

$$X=\bigcup_{y\in Y}f^{-1}\left(\left\{ y\right\} \right)$$ so $X$ is a countable union of countable sets, hence is countable.

Do you understand that? If not then have a look here.

edited Apr 13 '17 at 12:20

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answered Aug 17 '14 at 12:21

drhab

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First Course in Topology: Countable sets.

1 Answers1