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Math Help - Topologising RP2 using open sets in R#

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    Super Member Bernhard's Avatar
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    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
    Last edited by Bernhard; May 19th 2012 at 11:34 PM.
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