find close (duplicate) points + create index

Hello,

I have a table with one column --
"CREATE TABLE pointtable (point POINT)".
I'm trying to find duplicate entries, where two entries are considered
duplicates if they're within a radius of 1, meaning something like ~ "select
point from pointtable where distance between points <=1".
Obviously this is not SQL syntax. Is there a way to do this - search within a
column itself?

Also, I also tried to build an index on that column, but there's no operator
class for type point. How can I do that?

Thanks,
Elinor