Proving continuity with upper ideals in finite topological spaces

[Raziel2701]

Homework Statement

Here it is in all its glory: http://imgur.com/w08cp

Homework Equations

Here is a definition on what an upper ideal is: http://imgur.com/ZILjW
Here's what a finite topological space is: http://imgur.com/tBGTn

The Attempt at a Solution

From what I gather, I want to show that if $$B\in Ty$$, then the inverse image of B is an element of Tx.

From there I started by letting $$B\in Ty$$. Ty is a topology on Y and it is the set of upper ideals in $$(Y,\leq)$$. Thus for $$b\in B\in Ty$$, and $$y \in Y$$, $$b \leq y$$ so $$y \in Ty$$.

But I'm not seeing it, I don't know how one definition gets me to another step. Was I even in the right track?

 

[micromass]

Well, let's look at what you need to prove. Take $$B\in \mathcal{T}_Y$$.
You need to prove that $$f^{-1}(B)\in \mathcal{T}_X$$, thus you need to show that, given $$x\in f^{-1}(B)$$ and $$x\leq y$$, that $$y\in f^{-1}(B)$$...