Proving a function is continuous

[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)...