I think you want to cast your dt
to a date
and fix the format of your date literal:
SELECT *
FROM table
WHERE dt::date = '2011-01-01' -- This should be ISO-8601 format, YYYY-MM-DD
Or the standard version:
SELECT *
FROM table
WHERE CAST(dt AS DATE) = '2011-01-01' -- This should be ISO-8601 format, YYYY-MM-DD
The extract
function doesn't understand "date" and it returns a number.