Abstract
In this paper, a characterization is given for compact door spaces. We, also, deal with spaces
X such that a compactification
K
(
X
)
of
X is submaximal or door.
Let
X be a topological space and
K
(
X
)
be a compactification of
X.
We prove, here, that
K
(
X
)
is submaximal if and only if for each dense subset
D of
X, the following properties hold:
(i)
D is co-finite in
K
(
X
)
;
(ii)
for each
x
∈
K
(
X
)
∖
D
,
{
x
}
is closed.
If
X is a noncompact space, then we show that
K
(
X
)
is a door space if and only if
X is a discrete space and
K
(
X
)
is the one-point compactification of
X.