Vertical intervals as a basis for topology on ℝ²

[cragar]

Homework Statement

Show that the collection
$\{ \{a\}\times(b,c) \subset \mathbb{R^2} |a,b,c \in \mathbb{R} \}$
of vertical intervals in the plane is a basis for a topology on $\mathbb{R^2}$

The Attempt at a Solution

My question is just really about (a)X(b,c)
am I just basically letting a varying across the real line and then just pairing it up
with all the points on that line. So at each vertical interval
it will be open interval going up and down from point a and starting at b and going up vertically to c.

 

[Dick]

It's the open interval whose points all have first coordinate a and the second coordinate lies between b and c. So yes, if that's what you mean.

 

[nerever]

Hi. Could you explain your solution a bit more? I am stuck in the same problem..