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!)(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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

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

But I cannot see that this is the case.

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

Can someone please clarify this for me?

Peter

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# Topologising RP2 using open sets in R3

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