{b} is neither open nor closed
Example 5.7. The interval [0, 1] is compact in (R, Ostd). The one-element set containing (−1, 2) is a cover of [0, 1], but it is also a finite subcover and hence [0, 1] is compact from the definition.
It is clear that removing even one element from C will cause C to fail to be an open cover of R. Therefore, there is no finite subcover of C and hence, R is not compact.
Example
4.6
.
Let
M
=
{
a,b,c
}
and let
O
=
{
∅
,
{
a
}
,
{
a,b
}
,
{
a,b,c
}}
. Then
{
a
}
is open
but not closed,
{
b,c
}
is closed but not open, and
{
b
}
is neither open nor close
Example
4.6
.
Let
M
=
{
a,b,c
}
and let
O
=
{
∅
,
{
a
}
,
{
a,b
}
,
{
a,b,c
}}
. Then
{
a
}
is open
but not closed,
{
b,c
}
is closed but not open, and
{
b
}
is neither open nor close
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