In this paper, we prove that if a space
X
has a point-countable
c
n
-network, then the Pixley-Roy hyperspace
P
R
X
also has a point-countable
c
n
-network. If
X
is a regular space with a point-countable
c
k
-network, then so does the Pixley-Roy hyperspace
P
R
X
. Moreover, if
X
has a point-countable
s
p
-network (resp., strict Pytkeev network), then the Pixley–Roy hyperspace
P
R
2
X
also has a point-countable
s
p
-network (resp., strict Pytkeev network). On the other hand, we show that if the Pixley–Roy hyperspace
P
R
X
has a countable
c
n
-network (resp.,
s
p
-network and strict Pytkeev network), then so does
X
. By these results, we obtain that if the Pixley–Roy hyperspace
P
R
X
is a cosmic space (resp.,
P
0
-space, strict
P
0
-space, and stric
P
0
-space), then so is
X
. Furthermore, the Pixley-Roy hyperspace
P
R
n
S
2
is not a
k
-space for each
n
≥
2
.