Example:
Find a perfect set. Find a closed set that is not perfect. Find a
compact set that is not perfect. Find an unbounded closed set that
is not perfect. Find a closed set that is neither compact nor
perfect.
- A perfect set needs to be closed, such as the closed interval
[a, b]. In fact, every point in that interval [a, b] is
an accumulation point, so that the set [a, b] is a perfect set.
- The simplest closed set is a singleton { b }.The
element b in then set { b } is not an accumulation
point, so the set { b } is closed but not perfect.
- The set { b } from above is also compact, being closed
an bounded. Hence, it is compact but not perfect.
- The set
{-1}
[0, 
)
is closed, unbounded, but not perfect, because the element -1
is not an accumulation point of the set.
- The set
{-1}
[0, )
from above is closed, not perfect, and also not compact,
because it is unbounded.
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