I want to show that if $(X,\mathcal{T})$ is a metric topology. Take two disjoint subsets $A$ and $B$ such that forall $x\in A$ and $y\in B$ $d(x,y)\geq\delta$. Put the subspace topology on $A,B,A\cup B$. Prove that $A\cup B$ is homeomorphic to $A+B$ (where $(A+B)$ is the disjoint union topology)
My proof bsically involves using the obvious map: $f:A\cup B\rightarrow((A\times \{0\})\cup(B\times\{1\}))$ such that $f(x)=\begin{cases} (x,0) & \mbox{if}\ x\in A \\ (x,1) & \mbox{if} x\in B \end{cases}$
Then I need to apply the inverse of $f$ and $f^{-1}$ to open sets in the topology and show that they are open.
My idea for doing this is to note that as these are metric spaces they have a basis of unions of open balls.
Then if f and $f^{-1}$ map the open balls to open balls then that will be enough to show that they are continuous and so $f$ is a homeomorphism.
Now if I can show that the open balls in $A\cup B$ are all of the form (open balls in A) $\cup$ (open balls in B). That is there is no open ball in $A\cup B$ which is not the union of open balls in $A$ and open balls in $B$.
Then $f$ and $f^{-1}$ obviously map open balls to open balls.
My proof that if $B(a,\epsilon)$ is an open ball in $A\cup B$ then it is the union of an open ball in $A$ and an open ball $B$.:
Take an open ball $B(a,\epsilon)\subset (A\cup B)$ for $a\in A$ or $B$ such that $\exists x,y\in B(a,\epsilon)$ for $x\in A$ and $y\in B$.
Now consider $(B(a,\epsilon)\setminus B)\subset A$. We have to show that this is open (and so it is the union of open balls in $A$). Suppose that it was closed, then:
$\exists x\in (B(a,\epsilon)\setminus B)$ such that $\not\exists\ r>0:B(x,r)\subset A$.
However as $B(a,\epsilon)$ is open in $A\cup B$ then $\exists r>0:B(x,r)\subset (A\cup B)$
Now take $i$ such that $\frac{\delta}{i}
We can then use the same argument on $(B(a,\epsilon)\setminus A)$ to show that it is open. So any open ball in $A\cup B$ is a union of open balls in $A$ and open balls in $B$, where the open balls in $A$ and $B$ are open in $A\cup B$.
I this roughly correct (I've only given an outline of most of what I have done)
I have tagged this as homework as this was part of an assessment that I have already handed in and so is not really homework anymore but I'm not really looking for a complete solution to the problem if what I have done is incorrect just a small hint or something
Thanks very much for any help