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Show that there exists a continuous mapping of line $R$ with lower limit topology onto space $N$ with topology induced from euclidean topology on a line.

In the second space every singleton $\{n\}$ is a n open set, so I was thinking about creating a function for which $f^{−1}(\{n\})=[a,a+1)$, because it would be continuous which would mean that there exists such mapping. Does anyone have an idea how to do it precisely?

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You're actually almost there - you have exactly the right idea. Just pick $f(x)=\left \lfloor{x}\right \rfloor$, the floor function. Then, as you'd hoped, $f^{-1}(\{n\}) = [n,n+1)$. Thus the inverse image of any singleton is an open set. From here it's simple to show that the inverse image of every open set is open, and thus the function is continuous!

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