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Let $X$ be a topological space and $A \subseteq X$, $q:X \rightarrow X/A$ the quotient map. Let $Y$ be another topological space and $f:Y \rightarrow X/A$ continuous. Is there a $g:Y\rightarrow X$ continuous with $f=q\circ g$? Thank you.

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Some idea: The answer is no in general. As an example, consider $Y = X/A$ and $f: X/A \to X/A$ is the identity map. So you ask whether there is

$$g: X/A \to X$$

so that $id = q \circ g$?

For example, let $X = [0, 1]$ and $A = \{ 0, 1\}$. Can you find such a $g$?

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