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Math Help - Algebraic Topology - RP2 and R3

  1. #1
    Super Member Bernhard's Avatar
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    Algebraic Topology - RP2 and 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 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
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    Re: Algebraic Topology - RP2 and R3

    i think you are misunderstanding what he is doing. a given line L is not an open set in RP2. imagine it this way: suppose we have an open disk in R2. now add a third dimension, and "project" the disk into a cone (through the origin). open sets in RP2 will look like unions of these cones. in other words, open sets are "open bundles of lines".

    it might be easier to see what is happening in 2 dimensions. a "basic open set" in the real line is an open interval. now, draw a line parallel to the x-axis in R2, and consider the set of lines in R2 that pass through the open interval translated to that parallel line. what you get is essentially "two angle sectors", located on opposite sides of the unit circle (or any other circle centered at the origin, for that matter).

    it would be a mistake to think that all open sets in RP2 are "cone-shaped". it is better to think of them as "projections" of open sets in R2 through the origin (the projections look "flipped around" on the lower half-space determined by z = 0). something "impossible" happens at the origin (the shape of the projection isn't maintained), so it's best to avoid it.
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Algebraic Topology - RP2 and R3

    Thanks for that - most helpful

    So if we take any open set if R2 and project it through the origin - we get a cone-like structure (but not necessarily strictly a cone)

    Then we would be satisfying the key requirement - that is Crossley's statement that we "define a subset of RP2 to be open if the corresponding subset of R3 is open"

    Do you know of any good treatment of the topology of the projective line and plane - dealing with the homeomorphisms with the circle S1 and the sphere S2?

    Peter
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    Re: Algebraic Topology - RP2 and R3

    well RP2 isn't homeomorphic to the sphere S2, because the sphere is orientable, but the projective plane is not (imagine sewing up a disk "backwards" (with a twist)...you can't actually do this in 3 dimensions without self-intersection), it only has "one side" (because of the twist).

    if you read on in your text (around page 56 or so, i think), you will find an explicit description of the homeomorphism between RP1 and S1.

    you can, however, use the sphere to "visualize" the projective plane: what you do is identify the vector x on S2 with -x (both of these lie in the subspace defined by x, which is, of course, the very line that is taken to be the element of RP2 in your first post). this is actually a quotient map, which gives another way of "topologizing" RP2, as identification of pairs of "antipodal" open sets on the surface of the sphere: this is the "quotient topology".

    formally, we define an equivalence relation on S2, by x~y if y is in {x,-x}. clearly x~x, and if x~y, then either y = x, in which case x is in {y,-y}, or y = -x, in which case x = -y, so that x is in {y,-y}, thus y~x. transitivity is likewise fairly trivial. we then define a surjection q:S2→S2/~ = RP2 by q(x) = [x]. having done this, we define open sets in RP2 to be precisely those sets U such that q-1(U) is open in the (usual relative) topology on S2.

    that is, the quotient topology is the largest (that is:finest) topology that makes q continuous.

    it should be clear that these two ways of looking at RP2 are "equivalent":

    the map f:S2→RP2 given by f(x) = tx (where t is a real parameter) becomes a homeomorphism when x is replaced by [x]. the continuity of f is clear when we define the open sets of RP2 to be those sets of RP2 open in R3- {(0,0,0)}, for if U is such a set, then f-1(U) is open in the relative topology for S2, since it is the intersection of an open set in R3 with S2. furthermore, each line tx in RP2, corresponds to a unique pair {y,-y} = [y] in S2, namely, the pair of points {x/|x|, -x/|x|}, so f is bijective from S2/~ to RP2.

    finally, if we consider the map g:tx→[x/|x|], we see that for an open set V in S2/~, g-1(V) is the "cone" of lines through the origin that intersect V, which is open in R3-{(0,0,0)}, since it doesn't include "the edges of the cone". that is, S2/~ and RP2 are homeomorphic as topological spaces, so for all topological purposes "the same space".
    Thanks from Bernhard
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    Super Member Bernhard's Avatar
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    Re: Algebraic Topology - RP2 and R3

    Thanks so much for that post!!! I wish the textbooks were so clear!

    Skimming it I see that the post addresses the problems that motivated my post.

    I will work through the post carefully now and reflect on its implications

    Thanks again - it has really helped me to progress past a number of problems

    Peter
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