notes / Studies in Real and Complex Analysis / chapter 1: Metric and Topological Spaces / From my notebook
N7 Closure, Interior and Limit Points
Closed sets
Definition N7.1 (Closed set). is closed if is open.
So:
- and are closed;
- arbitrary intersections of closed sets are closed;
- finite unions of closed sets are closed.
Note. Most sets are neither open nor closed.
Closure
Definition N7.2 (Closure). For any subset, the closure of , , is the smallest closed set containing :
If is a disk without its edge, then is the disk with its edge:
Interior
Definition N7.3 (Interior). The interior of ( or ) is the largest open set contained in :
- there is an open such that .
- for all open , (i.e. ).
Boundary
Definition N7.4 (Boundary). The boundary of is or .
every open set in containing meets both and .
Dense sets
Definition N7.5 (Dense). A subset is dense in if , i.e. every nonempty open has .
Example N7.6. and are dense.
Limit points
Definition N7.7 (Limit point). is a limit point of a subset if for all open (called a neighborhood of , so some open set around ),
If is not a limit point, then is an isolated point of .
Recall for all open , .
(The two definitions match for .)
Corollary N7.9. is closed contains all its limit points.
Limit points and sequences
Limit of a sequence: in if for every neighborhood of there exists such that .
Facts. Let .
- If there is a sequence in such that , then .
- If there is a sequence in such that and for infinitely many , then is a limit point of .
Conversely, in a metric space: if (resp. is a limit point of ), then for all there exists (resp. with ), and then .