# Does surjective map induce surjective map on Hewitt real compactifications?

Let $$\beta X$$ be the Stone-Čech compactification and $$\upsilon X$$ be the Hewitt real compactification of a completely regular space $$X$$.

It is well known that any continuous surjective map $$f:X\rightarrow Y$$ induces surjective map $$\beta \left( f\right) :\beta X\rightarrow \beta Y$$. Similarly, Is the induced map $$\upsilon \left( f\right) :\upsilon X\rightarrow \upsilon Y$$ surjective?

So let $$f : X \to Y$$ be a continuous surjection, where $$X$$ is realcompact and $$Y$$ is not. Since $$X$$ is realcompact, $$\upsilon(X) = X$$, so $$\operatorname{im}(\upsilon(f)) = \operatorname{im}(f) = Y$$. But since $$Y$$ is not realcompact, $$\upsilon(Y) \supsetneq Y$$, and so $$\upsilon(f)$$ is not surjective.
• Actually, what I want to know whether the orbit map $\pi :X\rightarrow X/G$ induces surjective map between their real compactification spaces. The orbit space of a realcompact space is also realcompact. (R. Engelking, 3.11G.). Also, the orbit map is perfect open map. – Mehmet Onat Aug 9 '19 at 12:26