  We prove a general version of Radon's theorem for finite families $\mathcal
F$ of sets in zero-, one- and two-dimensional (pseudo)manifolds and graphs. The
only requirement is that for each $\mathcal G\subseteq\mathcal F$ the
intersection $\bigcap\mathcal G$ has bounded number of path-connected
components. As a consequence we obtain Helly's and Tverberg's theorems,
fractional and colorful Helly theorems, existence of weak $\varepsilon$-nets
and $(p,q)$-theorems.
  More precisely, if $\mathcal F$ is an intersection-closed family in a
topological space $X$ and $S\subseteq X$, we define its convex hull
$\text{conv}_{\mathcal F}(S)$ as the smallest set in $\mathcal F$ that contains
$S$. The Radon number $r(\mathcal F)$ is then the smallest integer $r$ such
that every set $S\subseteq X$ of size $r$ can be split into two disjoint
subsets with intersecting convex hulls.
  For every graph $G$ with $n$ vertices we construct a polynomial $p_G(x)$
satisfying the following three conditions. 1) If $n=1$, $p_G(x)=1$. 2) If
$n\geq 2$, the degree of $p_G(x)$ is at most $2n-3$. 3) If the graph $G$ does
not almost-embed into a topological space $X$, and $\mathcal F$ is a finite and
intersection-closed family of sets in $X$, such that each member of $\mathcal
F$ has at most $b$ path-connected components, then $r(\mathcal F)\leq p_G(b)$.
  In particular, if $\mathcal F$ is a family of sets in $\mathbb R^2$,
$r(\mathcal F)\leq r_{K_{3,3}}(b)\leq 2b^6 - 5b^5 + 10b^4 - 10b^3 + 9b^2 - 3b +
3$.
  The proof further refines the constrained chain map method introduced by
Goaoc, Pat\'ak, Pat\'akov\'a, Tancer and Wagner; and further developed by
Pat\'akov\'a. In the plane, we manage to overcome the use of hypergraph Ramsey
theorem and provide polynomial size bounds.
