1. ## continuous metric spaces

Show that

1. the projection map p : R^2 → R given by p(x; y) := x is continuous.
.

My proofs:

We want to show that $\displaystyle \forall$ $\displaystyle \epsilon$ > 0, $\displaystyle \exists$$\displaystyle \delta$>0 such that:

d(x,y)<$\displaystyle \delta$ $\displaystyle \Rightarrow$ d(f(x),f(y)) <$\displaystyle \epsilon$

(x1,y1) (x2,y2) are points in R^2

1) d(x1,x2) < $\displaystyle \epsilon$
|x1-x2| <$\displaystyle \epsilon$

We can define $\displaystyle \delta$ = the square root of ($\displaystyle \epsilon$^2 - (y2-y1)^2)

Since there exists a delta for any epsilon we choose, implies that p is continuous.

Is this correct...

2. Looks quite correct to me.