Complex projective space

Short definition
Complex projective space is defined as defining ingredient::projective space over a field of complex numbers $$\mathbb{C}$$, where the topological structure is induced from the structure of the field of complex numbers as a topological field.

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

As a set, we can think of it as the set of complex lines (which are planes in the real vector space sense) through the origin in $$\mathbb{C}^{n+1}$$. Using a Hermitian inner product on $$\mathbb{C}^{n+1}$$, it can also be identified with the set of hyperplanes of codimension 1 (i.e., $$n$$-dimensional complex linear subspaces) in $$\mathbb{C}^{n+1}$$.

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

Others
We can also consider the real projective space corresponding to any topological complex 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 $$\mathbb{C}^*$$.