Why is the subspace topology on RP^n difficult to grasp?

In summary, projective space (RP^n) is the set of all lines through the origin in R^3, and we can put a subspace topology on it by taking a subset of RP^n and the set of points in R^3 that are the union of these lines in the subset. However, the only possible topology we can get is the indiscrete topology because any neighborhood of zero contains a point on every line through the origin. This is due to the fact that in the definition of projective space, points are actually lines and a point on a line is not a point of projective space. To properly topologize RP^n, we need to modify the quotient topology to work on the unit sphere of R^n+
  • #1
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so first let's take RP^2. I have a little trouble grasping why we can put a subspace topology on it. So RP^2 is the set of all lines through the origin in R^3. So if we take some subset S of RP^2 and the if set of points in R^3 which is the union of these lines in S is open then we can say we take the intersection of S and the open subset of R^3 and get a subspace topology. However, apparently you can only get the indiscrete topology? the reason i read for this was because the union would contain 0 but you can't have an open ball around 0. Let's say you take a single line as your subset S. you can't put an open ball around any of the points right? you won't necessarily have some delta in the x, y, z directions? how come the argument says you can't put an open ball around 0, but you can put it around the other points?
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  • #2
The basic reason is that any neighborhood of zero contains a point on every line through the origin.
 
  • #3
so if we take some point P which is not 0, what would P's neighborhood consist of? and what would happen to the neighborhood when we take subsets of RP^n
 
  • #4
In the definition of projective space the lines through the origin are topologized by the set that they determine when intersecting the sphere. A set of lines is open if its intersection with the sphere is open in the sphere.

A point in projective space is a line. A point on this line is not a point of projective space. Zero is not a point nor is any other point on the line.
 
  • #5
The "correct" way to modify the quotient topology to work on RPn is to look at the unit sphere of Rn+1 and take the open sets on that, and match them with the sets of lines passing through the points of the open sets in Rn+1. This is the topology you're looking for in RPn
 

1. What is the definition of RP^n as a topological space?

RP^n is the real projective space, which is a topological space formed by identifying antipodal points on the n-dimensional sphere S^n. This means that any two points on the sphere that are diametrically opposite from each other are considered the same point in RP^n.

2. How is RP^n different from Euclidean space?

RP^n is a non-orientable space, meaning that it does not have a consistent notion of "left" or "right". This is because when identifying antipodal points on the sphere, we essentially "fold" the sphere onto itself, creating a Möbius strip-like structure. In contrast, Euclidean space is orientable and does not have this folding property.

3. What is the fundamental group of RP^n?

The fundamental group of RP^n is the cyclic group of order 2, denoted as Z/2Z. This means that any loop in RP^n can be continuously deformed into either itself or its reflection, but not into any other loop. Intuitively, this reflects the non-orientable nature of RP^n.

4. How does the topological structure of RP^n change as n increases?

As n increases, the dimension of RP^n increases and the space becomes more complex. For example, RP^1 is homeomorphic to a circle, RP^2 is homeomorphic to a sphere, and RP^3 is homeomorphic to a projective space, which can be visualized as a three-dimensional cross-cap or a three-dimensional Mobius strip. Beyond RP^3, it becomes more difficult to visualize, but the topological structure continues to be more complex.

5. What are some applications of RP^n in mathematics and science?

RP^n has applications in many areas of mathematics and science, including geometry, topology, and physics. In geometry, RP^n is used to study properties of projective spaces and projective geometry. In topology, RP^n is an important example of a non-orientable space and is used to understand the fundamental group and higher homotopy groups. In physics, RP^n is used in various theories, such as in string theory and quantum mechanics, to model non-orientable spaces and study their properties.

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