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- Mar 10, 2012

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**Definition.Identification topology:**Let $X$ be a topological space and let $\mathcal{P}$ be a partition of $X$. Consider a set $Y$ whose elements are the members of $\mathcal{P}$. Define the function $\pi : X \rightarrow Y$ which sends each point of $X$ to the subset of $\mathcal{P}$ containing it. A subset $O \subseteq Y$ is open in $Y$ if and only if $\pi ^{-1}(O)$ is open in $X$. So this gives a topology to $Y$ known as the identification topology on Y.

**Projective space description 1:**

Consider the unit circle $S^1$ in $\mathbb{R}^2$ and partition into subsets which contain exactly two points, the points being antipodal (at the opposite ends of the diameter). $P^2$ is the resulting identification space.

**Projective space description 2:**

Begin with $\mathbb{R}^2 - \{ 0 \}$ and identify two points if and only if they lie on the same straight line through the origin. The resulting identification space is $P^2$.

Show that the two descriptions of $P^2$ are equivalent.

I have no idea how to begin. Please help.