Let x R and let T U subset X 1 epsilon U or theta This i

Let x = R and let T = {U subset X : 1 epsilon U or = theta}. (This is an example of a particular point topology.) Describe the closed sets of X. The closed sets in this topology are the sets which do not contain I together with the set X. The only closed set containing the set {1} is X.

Solution

b) cl({1 , 2}) is the smallest closed set in R conaining 1 and 2 which is [1 , 2 ]

c) A subset of X is dense if for every x in X every open set containing x intersects the given set. Sinc every open set of X contains 1 it intersects with {1 , 2 }.thus {1 , 2 } is dense in X