I am studying basic projective and affine geometry and I've come across the notion of projective closure (by now, only in spaces obtained from K^n by projectivisation). It has been defined (somewhat vaguely) for an affine subspace described by a linear equation as the projective subspace described by the equation obtained homogenising the given one (i.e. multiplying the constant term for an additional variable). I probably haven't enough background to cope with more general definitions involving ideals and polynomials seen elsewhere, but I'd like to know if I can get more intuition by an analogy.
It's easy to see that if (x1, ..., xn) belongs of the initial subspace, then (x1 : ... : xn : 1) belongs to its projective closure: in this sense, projective closure "extends" the given subspace. I was wondering if this situation was interpretable by means of an analogy with topology, in which there are some closed set and the closure operator associates to every subset its closure, i.e. the smallest closed set containing it.
Are there somehow "closed" sets in the projective space, and the projective closure only associates to every set of the affine space the closure of the set obtained homogenising it? This could also lead to a formal definition generalising the one I was given: the homogenisation is described for every point, not only for linear equations, and the notion of closure would complete the definition.
Yes, there are topologies on affine and projective $n$-space that make your intuition true: these are the Zariski topologies on both spaces. Basically, a set is closed in the Zariski topology if it ''locally'' is the solution set to some system of algebraic equations, but this is imprecise because of course the notion of ''local'' needs a topology to be made sense of. Given the affine $n$-space $\mathbb{A}_K^n$ over $K$ and a chosen embedding $\mathbb{A}^n_K\hookrightarrow \mathbb{P}^n_K$ of it into projective $n$-space over $K$, and given a closed subspace $C$ of $\mathbb{A}^n_K$, it need not be true that $C$ is closed in $\mathbb{P}^n_K$. The closure of $C$ in $\mathbb{P}^n_K$ in the Zariski topology is precisely the projective closure as defined classically, if we choose the embedding $\mathbb{A}^n\hookrightarrow \mathbb{P}^n_K$ to be one of the standard ones (the closure depends of course on the choice of embedding).