Lines in algebraic subsets of infinite-dimensional projective spaces and connectedness (Q2467908)

The author introduces the notion of a line-connected projective subset in two steps. For a infinite-dimensional vector space \(V\) over a field \(K\) he considers a projective subset \(X\subset \mathbb P (V)\). Then he proves that \(X\) is line-connected in two steps if \(K\) is either an algebraically closed field or a finite field. In both cases this means that for all \(P,\, Q\in X\) there exist two lines \(D\subset \mathbb P (V)\) and \(R\subset \mathbb P (V)\) such that \(P\in D\), \(Q\in R\), \(D\cap R=\emptyset \) and \(D\cup R\subset X\).