Preimage Problem: Understand Closure & Metrics

[Ted123]

[PLAIN]http://img855.imageshack.us/img855/5949/metric.jpg

If X,Y are sets and f:X\to Y is a function with B\subset Y, the preimage is defined f^*(B) = \{x\in X : f(x) \in B\}.

If d_X, d_Y are metrics on X,Y, continuity of f can be characterised as follows:

The preimage of any open (resp. closed) set in (Y,d_Y) is open (resp. closed) in (X,d_X).

Hence, for example if we define f_1 (x,y) = x-y then f_1 is continuous and A_1 = f_1^*\left( (-\infty ,1] \right). Since (-\infty , 1] is closed, A_1 is closed.

Similarly for A_2 and A_3, but not sure about A_4. Can I write it in a way that makes it more obvious/easier to work with the preimage?

 

[HallsofIvy]

Left f(x,y)= x- y, a continuous function. The set of all (x, y) such that x- y\le 1 is the preimage, by that function, of the set \{z| z\le 1\}. Since that is a closed interval, it follows that A1 is a closed sert.

 

[Ted123]

I know that (A_1 is the one I proved already!) - it's [itexA_4[/tex] that I can't see how to do...