1. ## Topology. Metric Space.

Picture attached.

2. ## Re: Topology. Metric Space.

Originally Posted by vercammen
Picture attached.
You have posted an attachment without any comment whatsoever.
Do you expect someone to do the problem(s) for you?

Each of these follows from this theorem:
$\displaystyle (\forall x,~y)\left[|D(A,x)-D(A,y)|\le d(x,y)\right]$

The proof follows directly from the definition: $\displaystyle D(A,x)=\inf\{d(x,a):~a\in A\}~.$

Note that if $\displaystyle \delta>0$ then $\displaystyle \exists b\in A$ such that $\displaystyle D(A,x)\le d(x,b)<D(A,x)+\delta$.

Also note that $\displaystyle \forall a\in A$ we know $\displaystyle D(A,x)\le d(x,a)$.