Real projective space

Short definition
Real projective space is defined as defining ingredient::projective space over the defining ingredient::field of real numbers $$\R$$, with its topological structure induced by the structure of the field of real numbers as a defining ingredient::topological field.

Finite-dimensional
Real projective space of dimension $$n$$, denoted $$\R\mathbb{P}^n$$ or $$\mathbb{P}^n(\R)$$, is defined as the quotient space under the group action $$\R^{n+1} \setminus \{ 0 \}/\R^*$$ where $$\R^*$$ acts by scalar multiplication. It is equipped with the quotient topology.

As a set, we can think of it as the set of lines through the origin in $$\R^{n+1}$$. Using an inner product on $$\R^{n+1}$$, it can also be identified with the set of hyperplanes of codimension 1 (i.e., $$n$$-dimensional linear subspaces) in $$\R^{n+1}$$.

Countable-dimensional
This space, called countable-dimensional real projective space and denoted $$\R\mathbb{P}^\infty$$, is defined as the quotient space of the nonzero elements of a countable-dimensional real vector space (with the standard topology) over $$\R$$ by the action of $$\R^*$$ by scalar multiplication.

Others
We can also consider the real projective space corresponding to any topological real vector space, possibly infinite-dimensional, which is a real vector space equipped with a compatible topology. If the vector space is $$V$$, the projective space is defined as follows:we take $$V \setminus \{ 0 \}$$ with the subspace topology, and then put the quotient topology on its quotient under the action of $$\R^*$$.