# Math Help - filter

1. ## filter

Can anybody help me with this problem please?

f is mapping from X onto Y, F is filter on X, f(F) need not to be filter on Y.

I can't see no counterexample. Thank you very much

2. Originally Posted by sidi
Can anybody help me with this problem please?

f is mapping from X onto Y, F is filter on X, f(F) need not to be filter on Y.

I can't see no counterexample. Thank you very much
Let $\mathcal{F}$ be a filter on $X$. Clearly since $X\in\mathcal{F}$ then $Y=f(X)\in f\left(\mathcal{F}\right)$. Clearly $\varnothing\notin f\left(\mathcal{F}\right)$. Now, suppose that $f\left(E\right)\in f\left(\mathcal{F}\right)$ and let $G\subseteq f\left(E\right)$. Then, $f^{-1}(G)\subseteq E$ and thus $f^{-1}(G)\in \mathcal{F}$. Therefore, $f\left(f^{-1}(G)\right)=G\in\mathcal{F}$. So how many aximos are there left to violate?

3. really! Thanks a lot, I guess I am remade....thanks