I'm looking at families of curves in $\mathbb{P}^2$ (over $\mathbb{C}$), specifically the set $\mathcal{L}_d$ of projective curves defined by a homogeneous polynomial $P \in \mathbb{C}[x_0,x_1,x_2]$ of degree $d$ (possibly with repeated factors). In particular, I'm concerned with subsets of $\mathcal{L}_d$.
My specific question is this - if I have two subsets $S,S' \subseteq \mathcal{L}_d$ such that $S \subseteq S'$, with the further assumption that $S \simeq \mathbb{P}^N$ and $S' \simeq \mathbb{P}^N$ (for some $N$), is this enough to conclude that $S = S'$? I believe that if we were dealing with finite dimensional vector spaces this would be the case, and I'm aware that there is some link between projective and vector spaces, but I'm unsure what that link is and whether it's strong enough to allow this result to follow easily.
Now, I've given very specific details about $\mathcal{L}_d$ that probably aren't necessary to my basic query (but I've included them for context). Realistically, I think all I care about is whether the following holds:
Given sets $A,B$ with $B \subsetneq A$ and $A \simeq \mathbb{P}^k$ for some fixed $k$, if $B \simeq \mathbb{P}^{k'}$ for some $k'$ then $k' < k$.
More generally, is this some universal property that holds wherever the concept of dimension is well-defined? I'm just trying to wrap my head around what dimension actually means.