10.3. CONTINUITY OF A RESTRICTED MORPHISM
164
f

A
∈
C(
I
A
◦
µ
◦
I
A
;
ν
)
⇔
f

A
◦
I
A
◦
µ
◦
I
A
v
ν
◦
f

A
⇔
f
◦
I
A
◦
I
A
◦
µ
◦
I
A
v
ν
◦
f

A
⇔
f
◦
I
A
◦
µ
◦
I
A
v
ν
◦
f
◦
I
A
⇐
f
◦
I
A
◦
µ
v
ν
◦
f
⇐
f
◦
µ
v
ν
◦
f
⇔
f
∈
C(
µ
;
ν
)
.
f

A
∈
C
0
(
I
A
◦
µ
◦
I
A
;
ν
)
⇔
I
A
◦
µ
◦
I
A
v
(
f

A
)
†
◦
ν
◦
f

A
⇐
I
A
◦
µ
◦
I
A
v
(
f
◦
I
A
)
†
◦
ν
◦
f
◦
I
A
⇔
I
A
◦
µ
◦
I
A
v
I
A
◦
f
†
◦
ν
◦
f
◦
I
A
⇐
µ
v
f
†
◦
ν
◦
f
⇔
f
∈
C
0
(
µ
;
ν
)
.
f

A
∈
C
00
(
I
A
◦
µ
◦
I
A
;
ν
)
⇔
f

A
◦
I
A
◦
µ
◦
I
A
◦
(
f

A
)
†
v
ν
⇔
f
◦
I
A
◦
I
A
◦
µ
◦
I
A
◦
I
A
◦
f
†
v
ν
⇔
f
◦
I
A
◦
µ
◦
I
A
◦
f
†
v
ν
⇐
f
◦
µ
◦
f
†
v
ν
⇔
f
∈
C
00
(
µ
;
ν
)
.