### Ulla and Henriette live in **R**P^{∞}

Summarizing, the space **Σ** is a disjoint union of components - one component for each natural number (corresponding to the number of elements there might be in a finite set).
However, each component is in itself a fancy space (the ones corresponding to 0 and 1 are admittedly not that fancy, but from 2 on they become increasingly fascinating).

Take the component where {Ulla,
Henriette} lives. There we have non-trivial path

*f*: {Ulla, Henriette} -> {Ulla, Henriette}
given by cofusing Ulla and Henriette. This path is a loop, and it is not contractible (without letting go of the endpoints). On the other hand; if we use *f* **twice** we haven't done anything, so we insert a 2-cell so that cofusing twice is homotopic to not confusing anything.
Remember that in the picture, after identification there is just one vertex {Ulla, Henriette} and one edge *f*: the equality can have length zero.

So, {Ulla,Henriette} live in **R**P^{2}. However, it does not stop there; we have

*fff = f*, which - when you draw it - shows that {Ulla,Henriette} live in **R**P^{3} as well... and this continue to infinity:
{Ulla, Henriette} live in **R**P^{∞}

**R**P^{∞} is just another word for the *classifying space* BΣ_{2} of the group Σ_{2} with two elements.

**Σ** = 0 + BΣ_{1} + BΣ_{2} + BΣ_{3} + BΣ_{4} + ...

Negative sets

Bjørn Ian Dundas
2017-07-21 14:40:56 UTC