A weak fixed point property

The usual fixed point property can be interpreted in terms of non empty intersection of the graph of all maps with the graph of the identity map.

This motivates us to consider the following "weak fixed point property" by replacing the identity map with another continuous map:

A weak fixed point property $"P"$:

A topological space $X$ satisfies $P$ if there is a continuous map $F:X\to X$ such that for every continuous map $g:X\to X$ there exist a point $x_{0}$ such that $g(x_{0})=F(x_{0})$. That is: the graph of all maps intersect the graph of $F$.

We search for some examples (among topological space in particular manifolds) such that a space satisfies $P$ but does not satisfy the usual fixed point property. In particular do spheres, real or complex projective space satisfy $P$?

Your property is actually equivalent to the usual FPP. For, if a space $X$ does not satisfy FPP, that is there is a $\phi:X\to X$ with no fixed points, then, for any $F:X\to X$ the map $g:=\phi\circ F$ coincide with $F$ at no point: for any $x_0$, $g(x_0)=\phi(F(x_0))\neq F(x_0)$, so $X$ fails to satisfy property $P$.
• Thanks for your answer. An indirectly related question: Are there two continuous maps $F,g$ on $\mathbb{C}P^{2}$ where $F$ is surjective but their graphs do not intersect each other? are there two surjective maps on $\mathbb{C}P^{2}$ with non intersecting graphs? – Ali Taghavi Dec 19 '15 at 19:50