### 1.8.05

## a subset of the real line

take the interval (0,1)

throw away the middle closed interval of length 1/2, leaving two open intervals

throw away the middle closed interval of length 1/3 that of the whole interval, from each of those two

leaving four shorter open intervals

throw away the middle closed interval of 1/4 of the length from each of those

then the middle 1/5 (closed; including endpoints) from the eight open intervals you have left

and so on

for 1/6, 1/7 ...

the points you have left are an uncountable, nowhere dense set of measure zero which isn't open, but is

throw away the middle closed interval of length 1/2, leaving two open intervals

throw away the middle closed interval of length 1/3 that of the whole interval, from each of those two

leaving four shorter open intervals

throw away the middle closed interval of 1/4 of the length from each of those

then the middle 1/5 (closed; including endpoints) from the eight open intervals you have left

and so on

for 1/6, 1/7 ...

the points you have left are an uncountable, nowhere dense set of measure zero which isn't open, but is

*density open*. which means each point of the set is a density point of the set. which is the purpose of the exercise.