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 where, of course, consists of lines through the origin in .
We take a subset of i.e. a collection of lines in , and then take a union of these lines to get a subset of .
Crossley then defines a subset of to be open if the corresponding subset of 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 . (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 in his definition of openness, Crossley then asserts:
"Unions and intersections of correspond to unions and intersections of - {0} ..."
But I cannot see that this is the case.
If we consider two lines and 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 - {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 - {0} ??? ( again - see my diagram - topologising RP2 using open sets in R3 - attached) So the set is not open in - {0}?
Can someone please clarify this for me?
Peter