If
f
f
is a continuous surjection from a normal space
X
X
onto a regular space
Y
Y
, then there are a space
Z
Z
and a perfect map
b
f
:
Z
→
Y
bf:Z \to Y
extending
f
f
such that
X
⊂
Z
⊂
β
X
X \subset Z \subset \beta X
. If
f
f
is a continuous surjection from normal
X
X
onto Tychonov
Y
Y
and
β
X
∖
X
\beta X\backslash X
is sequential, then
Y
Y
is normal. More generally, if
f
f
is a continuous surjection from normal
X
X
onto regular
Y
Y
and
β
X
∖
X
\beta X\backslash X
has the property that countably compact subsets are closed (this property is called
C
C
-closed), then
Y
Y
is normal. There is an example of a normal space
X
X
such that
β
X
∖
X
\beta X\backslash X
is
C
C
-closed but not sequential. If
X
X
is normal and
β
X
∖
X
\beta X\backslash X
is first countable, then
β
X
∖
X
\beta X\backslash X
is locally compact.