Proving a function is continuous

  • Thread starter Raziel2701
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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 [tex]B\in Ty[/tex], then the inverse image of B is an element of Tx.

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

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?
 

Answers and Replies

  • #2
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Well, lets look at what you need to prove. Take [tex]B\in \mathcal{T}_Y[/tex].
You need to prove that [tex]f^{-1}(B)\in \mathcal{T}_X[/tex], thus you need to show that, given [tex]x\in f^{-1}(B)[/tex] and [tex]x\leq y[/tex], that [tex]y\in f^{-1}(B)[/tex]...
 

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