Who invented projective space $\mathbb{P}^n$ as an extension of the usual affine space $\mathbb{A}^n$?
Who was the first person to consider projective closure of plane affine algebraic curves (curves in $\mathbb{A}^2$)? Was it the same person?
Who invented projective space $\mathbb{P}^n$ as an extension of the usual affine space $\mathbb{A}^n$? Who was the first person to consider projective closure of plane affine algebraic curves (curves in $\mathbb{A}^2$)? Was it the same person? 


The idea of projective space goes back to the study of perspective in painting. The first formalization known is due to G. Desargues, with the book Brouillon Projet d'une atteinte aux événements des rencontres du Cône avec un Plan (Rough draft for an essay on the results of taking plane sections of a cone) published in 1639. There it was developed a geometry of incidence without parallel lines. It was very dense and difficult to read. Until XIX century the topic did not developed in full. Monge and Gergonne redeveloped it. Möbius introduced the homogeneous coordinates and Plücker also worked in these early developments. Steiner gave the first axiomatic (or synthetic) treatment. from there on, it playe a central role specially in the study of sets of solutions of polynomial equations. Today it makes one of the fundamental traits in modern algebraic geometry. But projective space considerations are present more or less implicitly also in topology, differential geometry, certain kind of differential equations and some descriptions of particle behavior in quantum mechanics. 


One should also mention Karl Georg Christian von Staudt (1798 – 1867), a German mathematician. His book "Geometrie der Lage (1847)" was a landmark in projective geometry. Staudt was the first to adopt a fully rigorous approach. Without exception his predecessors still spoke of distances, perpendiculars, angles and other entities that play no role in projective geometry. 


If the question concerns projective space specifically rather than projective geometry in a broader sense, then the answer would have to be JeanVictor Poncelet (1788 – 1867). Desargues already introduced the notion of a point at infinity, but I believe Poncelet was the first to consider a LINE at infinity. 

