(Problem 3-7 John Lee’s Introduction to Topological Manifolds) Let $X$ be the set $\Bbb R^2$ with the topology generated by nonempty collections of sets of the form $$I_{abc}=:\{(c,y)\mid a<y<b \land a,b,c\in\Bbb R\}.$$ Show that $X$ is homeomorphic to $\Bbb R_d\times \Bbb R$, where $\Bbb R_d$ is the set of real numbers endowed with the discrete topology.
Thoughts: If we can construct a homeomorphism $\varphi: X\to\Bbb R_d\times \Bbb R$, then our condition is proved. So, in that vain, suppose that I had a suitable map $\varphi$ and an open subset $U\subseteq \Bbb R_d\times\Bbb R$. By definition, $U$ can be written as the arbitrary union of sets of the form $$\mathcal{O}= V\times W,$$ where $V\subseteq\Bbb R_d \land W\subseteq\Bbb R$ are open sets in their respective spaces. If we consider a sequence $\mathcal{O}_{\alpha}$ of such sets indexed by $A$, we get $$\varphi^{-1}\left(\bigcup_{\alpha\in A}\mathcal{O}_{\alpha}\right)= \bigcup_{\alpha\in A} \varphi^{-1}(\mathcal{O}_{\alpha}) =\bigcup_{\alpha\in A}\varphi^{-1}(V_{\alpha})\times\varphi^{-1}(W_{\alpha}).$$ Thus, if we know that $\varphi^{-1}(V_{\alpha})\times \varphi^{-1}(W_{\alpha})$ is a nonempty collection of sets of the form $I_{abc}$, then we can conclude that $\varphi$ is continuous. Moreover, we can construct the map $\varphi$ with this constraint in mind, but I cannot think of a map for which this holds.
Next, we consider the inverse map $\varphi^{-1}:\Bbb R_d\times\Bbb R\to X$ and an open subset $U\subseteq X$. By hypothesis, we know that $U$ is generated by nonempty collections of sets of the form $I_{abc}$. From here, I am unsure as to how I should proceed.
I would appreciate any hints as to how I can
- Glean information about $\varphi$ from my first argument
- Find suitable map such that $\varphi^{-1}$ is continuous.
Am I missing something obvious? Thanks in advance.
The identity map id, is a homeorphism.
Show the given base for X is also a base for Y = $R_d×R$.
Use that to show the bijection id is open and continuous.
Hence id is a homeomorphism between two identical spaces,