I am reading Martin Crossley's book - Essential Topology - basically to get an understanding of Topology and then to build a knowledge of Algebraic Topology! (That is the aim, anyway!)

On page 27, Example 3.33 (see attachment) Crossley is explaining the toplogising of $\displaystyle \mathbb{R} P^2 $ where, of course, $\displaystyle \mathbb{R} P^2 $ consists of lines through the origin in $\displaystyle \mathbb {R}^3 $.

We take a subset of $\displaystyle \mathbb{R} P^2 $ i.e. a collection of lines in $\displaystyle \mathbb {R}^3 $, and then take a union of these lines to get a subset of $\displaystyle \mathbb {R}^3 $.

Crossley then defines a subset of $\displaystyle \mathbb{R} P^2$ to be open if the corresponding subset of $\displaystyle \mathbb {R}^3 $ is open.

Crossley then argues that there is a special problem with the origin, presumably because the intersection of a number of lines through the origin is the origin itself alone and this is not an open set in $\displaystyle \mathbb {R}^3 $. (in a toplological space finite intersections of open sets must be open)[Is this reasoning correct?]After resolving this problem by omitting the origin from $\displaystyle \mathbb {R}^3 $ in his definition of openness, Crossley then asserts:

"Unions and intersections of $\displaystyle \mathbb{R} P^2 $ correspond to unions and intersections of $\displaystyle \mathbb {R}^3 $ - {0} ..."

But I cannot see that this is the case.

If we consider two lines $\displaystyle l_1 $ and $\displaystyle l_2 $ passing through the origin(see my diagram - topologising RP2 using open sets in R3 - attached)then the union of these is supposed to be an open set in $\displaystyle \mathbb {R}^3 $ - {0} . But surely this would only be the case if we consider a complete cone of lines through the origin. With two lines - take a point x on one of them - then surely there is no open ball around this point in $\displaystyle \mathbb {R}^3 $ - {0} ???( again - see my diagram - topologising RP2 using open sets in R3 - attached)So the set is not open in $\displaystyle \mathbb {R}^3 $ - {0}?

Can someone please clarify this for me?

Peter