FactChecker said:
All true, but you should probably mention what properties of the square root function you are using in the last two inequalities of the triangle property proof. There are places where you are applying a property of the square root function that you should mention (and maybe prove).
Not sure what name this is called, but it seemed clear from
##\sqrt{d(x,y)+d(y,z)}\leq\sqrt{d(x,y)+d(y,z)+2\sqrt{d(x,y)d(y,z)}}=\sqrt{d(x,y)}+\sqrt{d(y,z)}##
pasmith said:
I think the last part is best done by noting that for every x_0 \in X and every r \geq 0 we have <br />
\{ x \in X : d(x,x_0) < r \} = \{ x \in X: d'(x,x_0) = \sqrt{d(x,x_0)} < \sqrt{r}\}. So if a set contains a d-open ball of each of its points then it contains a d'-open ball of each of its points and vice-versa.
I agree with you. My main difficulty was explaining how ##d(x,y)<r## is the same as ##d'(x,y)<\sqrt{r}##
PeroK said:
I would prefer something more explicit. Showing that ##d'## is a metric by almost never writing ##d'## seems not quite right to me. I feel I'm having to fill in too many blanks.
Also, for the open sets question, I'd like to see something more formulaic:
Let ##U## be open under ##d##, then ... ##U## is open under ##d'##.
Let ##U## be open under ##d'##, then ... ##U## is open under ##d'##.
##d'## is a metric on ##X## because it satisfies the axioms of metrics:
Identity of indiscernibles:
##x=y\Longleftrightarrow d(x,y)=0\Longleftrightarrow \sqrt{d(x,y)}=\sqrt{0}\Longleftrightarrow d'(x,y)=0##
Symmetry: ##d(x,y)=d(y,x)\Longrightarrow \sqrt{d(x,y)}=\sqrt{d(y,x)}\Longleftrightarrow d'(x,y)=d'(y,x)##
Triangle inequality: ##d(x,z)\leq d(x,y)+d(y,z)\Longrightarrow \sqrt{d(x,z)}\leq \sqrt{d(x,y)+d(y,z)}\leq \sqrt{d(x,y)}+\sqrt{d(y,z)}\Longleftrightarrow d'(x,z)\leq d'(x,y)+d'(y,z)## (because the square root of the sum is less than equal to the sum of the square roots)Let ##U## be open under ##d##, then for all ##x## in ##U##, there is an ##\epsilon>0## such that any ##y## that satisfies ##d(x,y)< \epsilon## is in the open set ##U## for ##d##. Define ##\delta## as ##\sqrt{\epsilon}##, then any ##y## that satisfies ##d(x,y)<\epsilon## also satisfies ##d'(x,y)< \delta##, and is in the open set ##U## for ##d##.
Let ##U## be open under ##d'##, then for all ##x## in ##U##, there is a ##\delta>0## such that any ##y## that satisfies ##d(x,y)< \delta## is in the open set ##U## for ##d'##. Define ##\epsilon## as ##\delta^2##, then any ##y## satisfies ##d'(x,y)< \delta## also satisfies ##d(x,y)<\epsilon##, and is in the open set ##U## for ##d##.