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...