Two problems related to continuity of a metric from Munkres' topology book

Solution 1:

Some hints:

• Use the following definition of continuity: a function $f:A\to B$ is continuous at $x\in A$ if for every open neighbourhood $U\subseteq B$ of $f(x)$ there exists an open neighbourhood $V\subseteq A$ of $x$ so that $f(V)\subseteq U$. So in this case, start by fixing $(x,y)\in X\times X$ and let $B(d(x,y),r)\subseteq \mathbb R$ be an open disc centered at $d(x,y)$ with radius $r>0$. Show that for all $(u,v)\in B(x,\frac{r}{2})\times B(x,\frac{r}{2})$ we have $$|d(u,v)-d(x,y)|<r$$ by using the triangle inequality, which would imply $d(B(x,\frac{r}{2})\times B(x,\frac{r}{2}))\subseteq B(d(x,y),r)$ and thus the continuity of $d$.

• Fix $x\in X$ and note that $y\mapsto d(x,y)$ is a continuous function $X'\to \mathbb R$. Then express each ball $B_{d}(x,r)=\{y\in X:d(x,y)<r\}$ as the preimage of an open set under a continuous function in the topology $X'$. Since each open ball in $X$ is open in $X'$, then the topology of $X'$ is finer than the topology of $X$.