$\Bbb R \times \Bbb R$ with a nonstandard topology Let $\tau$ the topology on $\Bbb R \times \Bbb R$ generated by the collection of lines $y= 2x+k$ with $k\in \Bbb R$ (in the sense that the line $\{y=2x+k\}$ is a basic element for $\tau$) .
Find $\tau_1, \tau_2$ such that $(\Bbb R \times \Bbb R,\tau)$ is homeomorphic to $(\Bbb R, \tau_1) \times (\Bbb R,\tau_2)$.
EDIT: Corrected spelling. Sorry for the inconvenience.

The problem is that I can't write rigorously what I think on this:
Since an open set can always be written as a union of basics elements, open sets in this "line-topology" are bands, or union of bands. Looking at the Y-axis (as an example) the basic element $(a,b)$ in the standard topology is reflected in the band $y=2x+k$, with $k\in (a,b)$ and vice versa. So, $\tau_2$ should be the standard topology.
On the other side, I know that if $f\colon X \to Y$ is a continuous map, then $X$ is homeomorphic to the graph of $f$. Applying this to the exercise, $\tau_1$ should be the trivial topology, $\tau_1 = \{ \emptyset, \Bbb R\}$
So, as always, any help will be appreciated.  
 A: For each $k\in\Bbb R$ let $L_k$ be the line whose equation is $y=2x+k$; I interpret the statement that $\tau$ is defined by these sets to mean that $\{L_k:k\in\Bbb R\}$ is a base for $\tau$. Thus, the set of points on any family of these lines is open in $\tau$, not just bands. That is, for any $A\subseteq\Bbb R$, $\bigcup_{k\in A}L_k\in\tau$.
Now consider one of these basic open sets, say $L_k$. Does it have any open subsets? Only $\varnothing$ and $L_k$ itself, because $L_k\cap L_j=\varnothing$ when $j\ne k$. Thus, the relative topology on $L_k$ induced by $\tau$ is the indiscrete (trivial) topology. Now there’s obvious bijection between $\Bbb R$ and $L_k$, given by the map
$$f_k:\Bbb R\to L_k:x\mapsto\langle x,2x+k\rangle\;;$$
this map will be a homeomorphism if and only if we give $\Bbb R$ the indiscrete topology $\tau_2=\{\varnothing,\Bbb R\}$ as well. Thus, as a subspace of $\langle\Bbb R\times\Bbb R,\tau\rangle$ each $L_k$ is a copy of $\langle\Bbb R,\tau_2\rangle$, and $\langle\Bbb R\times\Bbb R,\tau\rangle$ ‘looks like’ a bunch of copies of $\langle\Bbb R,\tau_2\rangle$ side by side. How many copies? One for each $k\in\Bbb R$. This gives us a map from $\Bbb R\times\Bbb R$ to the space $\langle\Bbb R\times\Bbb R,\tau\rangle$ that sends the pair $\langle k,x\rangle$ to the point $f_k(x)$ on the line $L_x$:
$$h:\Bbb R\times\Bbb R\to\Bbb R\times\Bbb R:\langle k,x\rangle\mapsto f_k(x)=\langle x,2x+k\rangle\;.$$
Note that $h$ takes $\{k\}\times\Bbb R$ to $L_k$.


*

*Show that $h$ is a bijection.  

*Find a topology $\tau_1$ on $\Bbb R$ so that $h$ is a homeomorphism from $\langle\Bbb R,\tau_1\rangle\times\langle\Bbb R,\tau_2\rangle$ to $\langle\Bbb R\times\Bbb R,\tau\rangle$.

