I have defined the set $H_{a,b,c} = \{(x,y) \in \mathbb{R}^2 \vert x = at + b, y = at + c, 0 < t < 1\}$.
Then I let $\mathcal{B} = \{H_{a,b,c} \vert a,b,c \in \mathbb{R}\}$. I want to study the relationship between the topology induced by the base $\mathcal{B}$ and the euclidean topology.
Now, proving that $\mathcal{B}$ induces a topology is rather straightforward, but I'm struggling to understand the relationship with the euclidean topology. I mean, to me an element of the base is the straight line $y = x -b +c$ between 0 < t < 1 so actually it is a segment. But where do I draw the line on the plane in this form if there is not anymore in the equation?
Besides, to me a segment does not have a relationship with an element of the base of the euclidean topology, because in the euclidean topology I either have balls or rectangles.
Any hints?