Continous Deformation of Hypersurfaces

Continous Deformation of Hypersurfaces

I'm reading Katz "Enumerative Geometry and String Theory". He shows that a
degree $d$ hypersurface has cohomology class $dH $(where $H$ is the class
of a hyperplane) by the following argument (p.80, I'm paraphrasing a bit):
...we can continuously deform a degree d hypersurface (given by F=0) to a
union of d >hyperplanes by the equations: $$ G_t := tF(x) + (1-t)
\Pi_{i=1}^{d} l_i (x) = 0 $$ where $ l_i $ are homogeneous linear forms.
I've studied the topic of intersection theory a bit already, so my
question isn't about the result necessarily, but more about his argument.
Specifically, what does he mean by continuous here (i.e. continous in what
topology)? For example, if we are in the the plane, and we take $d=2$, he
is saying we can continuously deform a circle into a pair of lines. I'm
pretty sure that deformation isn't continuous, though (or maybe it is in
the complex plane?)