## Topologising RP2 using open sets in R3

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 $\mathbb{R} P^2$ where, of course, $\mathbb{R} P^2$ consists of lines through the origin in $\mathbb {R}^3$.

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

Crossley then defines a subset of $\mathbb{R} P^2$ to be open if the corresponding subset of $\mathbb {R}^3$ 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 $\mathbb {R}^3$. (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 $\mathbb {R}^3$ in his definition of openness, Crossley then asserts:

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

But I cannot see that this is the case.

If we consider two lines $l_1$ and $l_2$ 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 $\mathbb {R}^3$ - {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 $\mathbb {R}^3$ - {0} ??? ( again - see my diagram - topologising RP2 using open sets in R3 - attached) So the set is not open in $\mathbb {R}^3$ - {0}?

Can someone please clarify this for me?

Peter

 PhysOrg.com science news on PhysOrg.com >> Heat-related deaths in Manhattan projected to rise>> Dire outlook despite global warming 'pause': study>> Sea level influenced tropical climate during the last ice age
 Recognitions: Science Advisor There is a natural map from 3 space minus the origin onto the projective plane. A point is mapped to the line through the origin that contains it. the topology of the projective plane is just the quotient topology under this map. Inverse images of open sets therefore are open sets in 3 space minus the origin. In fact, a set is open in the plane only if its inverse image is open. A basis for the topology of projective space is the projections of open cones of lines through the origin. (The origin is removed from these cones to give an open set in 3 space. )

Thanks for the help

So that means that I cannot simply take two lines (points) in $\mathbb{R} P^2$ as an open set because the corresponding set in $\mathbb{R}^3$ is not open - as in the attached digram.

Only cones of lines in $\mathbb{R} P^2$ are open.

But this seems to contradict what Crossley says - see attachement of Crossley page 27.

Peter
Attached Files
 Toplogising RP2 - using open sets in R3.pdf (20.8 KB, 10 views) Essentail Topology by Martin Crossley - Page 27.pdf (48.0 KB, 6 views)

Recognitions:
 Quote by Math Amateur Thanks for the help So that means that I cannot simply take two lines (points) in $\mathbb{R} P^2$ as an open set because the corresponding set in $\mathbb{R}^3$ is not open - as in the attached digram. Only cones of lines in $\mathbb{R} P^2$ are open. But this seems to contradict what Crossley says - see attachement of Crossley page 27. Peter