Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a *bijective, continuous* map $\phi:K \to K$ which satisfies the condition
$$
\hat u + \hat v \supset \hat w \implies \phi(\hat u) + \phi(\hat v) \supset \phi(\hat w) \tag{1}
$$
for all $\hat u,\hat v,\hat w \in P^{n-1}(\Bbb C)$. Here, I am interpreting $\hat u \in P^{n-1}(\Bbb C)$ to be a $1$-dimensional subspace of $\Bbb C^n$, so that $\hat u + \hat v$ is the smallest subspace containing both $\hat u$ and $\hat v$.


The question is as follows: can I guarantee that $\phi$ will have an extension $\Phi:X \to X$ which satisfies (1) for all $\hat u,\hat v, \hat w \in X$? 


The motivation here is that, by the fundamental theorem of projective geometry, such a map $\Phi$ is necessarily induced by a semilinear map; it would be nice if I could guarantee that $\phi$ is induced by such a map. If anyone could point me in the direction of a theorem or reference that might be useful here, I'd appreciate it. 




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*Note: I've done my best to make this question self-contained and concise, and have therefore left out background that I felt was extraneous. Feel free to comment if you feel that I should provide further context or clarification*



Extending a continuous map over projective space
Your condition (1) means: if $\hat{u}$, $\hat{v}$, and $\hat{w}$ are linearly dependent, then so are $\widehat{\varphi(u)}$, $\widehat{\varphi(v)}$, and $\widehat{\varphi(w)}$. So $\varphi$ preserves linear dependence of $3$ vectors. But it doesn't detect linear dependence among $4$ or more vectors.


The other answers have given counterexamples where $K$ is a finite set of points. Basically, it's not hard to construct finite sets of points where no $3$ are linearly dependent, but there is a dependence among some $4$ or more of them.


I'll give a counterexample where $K$ is connected. Let $K$ be a rational normal curve, that is, the curve parametrized by $[s^{n-1} : s^{n-2} t : s^{n-3} t^2 : \dotsb : t^{n-1} ]$ in homogeneous coordinates on $X = \mathbb{P}^{n-1}$. Now $K$ doesn't have any trisecant lines, i.e., there are no $3$ points $\hat{u},\hat{v},\hat{w} \in K$ that are collinear in projective space (equivalently, $u,v,w \in \mathbb{C}^n$ are linearly dependent). So condition (1) is vacuously true. Then take $\phi$ to be any old map on $K$; then $\varphi$ satisfies (1). But if $\phi$ isn't semilinear then it doesn't extend to $X$.


