  The long-time behavior is one of the most fundamental properties of dynamical
systems. Poincar\'e studied the Poisson stability to capture the property of
whether points return arbitrarily near the initial positions. Birkhoff studied
the concept of recurrent points. Hilbert introduced distal property to describe
a rigid group of motions. We show that Poisson stability, recurrence, and
distal property of flows on surfaces are topological properties. In fact, a
flow on a connected compact surface is Poisson stable (resp. recurrent) if and
only if the Kolmogorov quotient of the orbit space satisfies $T_{1}$ (resp.
$T_{1/2}$) separation axiom. Moreover, Poisson stability for such flows is
equivalent to distal property. In addition, $T_2$ separation axiom corresponds
to the isometric property. In addition, we construct ``Lakes of Wada continua''
which are the singular point sets of recurrent non-Poisson-stable flows and
Poisson stable distal non-equicontinuous flows on surfaces.
