Exercise: Let $S$ be the collection of all straight lines in the plane which are parallel to the $X$-axis. If $S$ is a sub basis for a topology $\tau$ on $\mathbb{R}^2$, describe the open sets in $(\mathbb{R}^2,\tau)$.
My attempt: If the lines were not parallel to the $X$-axis, every point (a,b) would be the intersection of two lines with different slope, then we would have the discrete topology.
Since the slope is the same:
If $(x,y)\in A$, then $(x,y)\in A\forall x\in \mathbb{R}$. So the open set $\mathbb{R}\times B$ where $B\subset\mathbb{R}$.
Questions:
1) How do I proceed to describe the open sets?
2)How do I prove $\mathbb{B}\times\mathbb{R}$ is open?
Thanks in advance!
Best Answer
The subbase generated by the lines $L_y:= \mathbb{R} \times \{y\}$ generates a base of its topology by taking all finite intersections. But all of these are empty (for two different sets) so the subbase $\{L_y: y \in \mathbb{R}\}$ is already a base.
This means that all open sets are unions of these sets if $O = \bigcup \{L_y: y \in B\}$ for some set $B$ then in fact $O = \mathbb{R} \times B$. So the topology generated is
$$\mathcal{T}= \{ \mathbb{R} \times B: B \subseteq \mathbb{R}\}$$ where $B$ ranges over all subsets of $\mathbb{R}$.
It follows that no set of the form $B \times \mathbb{R}$ is open, unless $B = \mathbb{R}$ or $B = \emptyset$, when we get the open set $\mathbb{R}^2$ and $\emptyset$ respectively.