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Tietze's extension theorem says:

''If $A$ is a closed subset of $X$ a normal space, and $f:A\to \mathbb{R}$ continuous, then we can extend $f$ to a continuous function $g:X\to \mathbb{R}$."

I know that it can be generalized by changing $\mathbb{R}$ to closed subsets of $\mathbb{R}^n$. But, can we find further generalizations by changing $\mathbb{R}$ to a more general space $Y$. To be precise, I'm looking for generalizations for an arbitrary polish space $Y$, and if it's worth it, $X$ is a zero dimensional polish space.

The thing is, I want continuous functions, but I really only care how they behave in a closed set. For any particular case, I can use a nice Cantor scheme and get the function, but I don't want to keep repeating me.

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    @MartinSleziak: My sets are already $G_\delta$, so Lavrentieff's wouldn't really help. In fact, I only really need $X$ to be any zero-dimensional Polish space, so I guess I could only take $X$ to be my set $A$ and call it a day, but I was striving for a more elegant approach and have my function from the whole Cantor space.2012-06-10

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