Recall that a topological space $X$ has the *fixed point property* (FPP) if any continuous function $f: X\to X$ has a fixed point.


Is the notion of FPP for topological spaces an absolute notion? More precisely:


**Question.** Is it consistent that a topological space $X$ has the FPP in $V$, but it does not have the FPP in some (cardinal preserving) generic extension of $V$?



Is the notion of fixed point property for topological spaces an absolute notion?
The answer is that the FPP is not absolute, and indeed, even the unit interval loses the FPP in a forcing extension. The unit interval famously has the FPP, but I claim that in any forcing extension having a new real, such as the forcing extension $V[c]$ obtained by adding a Cohen real, which preserves cardinals, the ground model unit interval no longer has the FPP. 


To see this, let $X=I^V$ be the unit interval of $V$, considered as a topological space in $V[c]$. Let $a\_n\to c$ be an increasing sequence of rational numbers converging to $c$ from below, and $b\_n\to c$ from above. Let $f:X\to X$ be the piece-wise linear increasing function that takes the interval $[a\_n,a\_{n+1}]\to [a\_{n+1},a\_{n+2}]$ and similarly $[b\_{n+1},b\_n]\to [b\_{n+2},b\_{n+1}]$. This function takes ground-model reals to ground-model reals, but below $c$, it lies above the diagonal and above $c$, it lies below the diagonal. So it has no fixed point in $X$. (Meanwhile, it has a natural extension to the unit interval in $V[c]$, having fixed point $c$.)


