# How to construct a continuous map such that the preimage of a particular non-open set is not open?

EDIT

This might be a more clear problem statement:

1. We have an arbitrary topological space $$Y$$ and an non-open set $$B$$ in $$Y$$. How to find a topological space $$X$$ and a continuous map $$F:X\rightarrow Y$$ such that $$F^{-1}(B)$$ in not open in X?

2. We have an arbitrary topological space $$Y$$ and an non-open set $$B$$ in $$Y$$. But now an arbitrary topological space $$X$$ is given, such that there exists a continuous map $$F:X\rightarrow Y$$. How to construct a continuous map $$G$$ such that $$G^{-1}(B)$$ is not open in $$X$$?

Original questions

We have two topological spaces $$X$$ and $$Y$$.

If $$B$$ is not open in $$Y$$, is there a continuous map $$F:X \rightarrow Y$$ such that $$F^{-1}(B)$$ is not open in $$X$$? How to construct such a map?

Equivelently, if for $$\forall$$ continuous function $$F:X\rightarrow Y$$ we have that $$F^{-1}(B)$$ is open in $$X$$, is $$B$$ open in $$Y$$? Can you give a counter example?

Under the discrete topology on a set $$X$$, any subset of $$X$$ is open and, for any topological space $$Y$$, any function $$F: X \to Y$$ is continuous. For a specific counter-example to both your conjectures, take $$X$$ to be $$\Bbb{R}$$ with the discrete topology and $$Y$$ to be $$\Bbb{R}$$ with the usual topology and $$F : X \to Y$$ to be the identity function ($$F(x) = x$$).
1. Take $$X = Y$$ and $$F : X \to Y$$ to be the identity map.
2. There is no such construction that will work for any $$X$$. First note that the assumption that there exists a continuous function $$F : X \to Y$$ is not helpful, since it is true for any $$X$$ and $$Y$$, because constant functions are continuous. If $$X$$ is discrete and $$G : X \to Y$$ is any function, then $$G^{-1}(B)$$ is open for any $$B$$.
• Thank you for the example. What if the set $X$ and $Y$ and the topologies on them are arbitrarily chosen as long as there exists a continuous map $F: X \rightarrow Y$? Is there a general way to construct a continuous map such that the preimage of a given non-open set $B$ in $Y$ is not open in $X$?