---- > [!definition] Definition. ([[real projective space]]) > The **real projective space** $\mathbb{R}P^{n}$ is defined to be the [[quotient topology|quotient space]] of $\mathbb{R}^{n+1} \cut \{ \b 0 \}$ by the [[equivalence relation]] $\b x \sim \alpha \b x \text{ for all } \alpha \in \mathbb{R}_{\neq 0} \text{ and } x \in \mathbb{R}^{n+1}_{\neq \b 0}.$ > \ > It is the real [[Grassmannian]] $\text{Gr}(k,n)$ in the special case $k=1$. Points in $\mathbb{R}P^{n}$ are most often expressed in [[projective space as a smooth manifold|homogeneous coordinates]]. > [!generalization] > - [[projective space]] ^generalization ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```