# Closed Set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation.

### Other articles related to "sets, closed set, closed sets, closed, set":

Alexandrov Topology - Characterizations of Alexandrov Topologies
... An arbitrary intersection of open sets in X is open ... Closed set characterization ... An arbitrary union of closed sets in X is closed ...
Closure (mathematics)
... A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set ... For example, the real numbers are closed under subtraction, but the natural numbers are not 3 and 8 are both natural numbers, but the result of 3 − 8 is not a natural number ... Another example is the set containing only the number zero, which is a closed set under multiplication ...
Spalding–Rigdon Theory Of Book Of Mormon Authorship - Computer Analysis
... of NSC methodology" led to "misleading results" by Jockers et al because they had used a closed set of 7 authors for their study. 2009) showed that an open set of candidate authors "produced dramatically different results from a closed-set NSC analysis." The Schaalje peer-reviewed. 2008 study by noting numerous problems, including the closed set analysis that forced the choosing of a winner while excluding the possibility that an author outside the closed set could be selected ...
Glossary Of Topology - C
... Clopen set A set is clopen if it is both open and closed ... Closed ball If (M, d) is a metric space, a closed ball is a set of the form D(x r) = {y in M d(x, y) ≤ r}, where x is in M and r is a positive real number, the radius ... A closed ball of radius r is a closed r-ball ...
... In point set topology, a set A is closed if it contains all its boundary points ... The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces ... An alternative characterization of closed sets is available via sequences and nets ...

### Famous quotes containing the words set and/or closed:

This ferry was as busy as a beaver dam, and all the world seemed anxious to get across the Merrimack River at this particular point, waiting to get set over,—children with their two cents done up in paper, jail-birds broke lose and constable with warrant, travelers from distant lands to distant lands, men and women to whom the Merrimack River was a bar.
Henry David Thoreau (1817–1862)

Don: Why are they closed? They’re all closed, every one of them.
Pawnbroker: Sure they are. It’s Yom Kippur.
Don: It’s what?
Pawnbroker: It’s Yom Kippur, a Jewish holiday.
Don: It is? So what about Kelly’s and Gallagher’s?
Pawnbroker: They’re closed, too. We’ve got an agreement. They keep closed on Yom Kippur and we don’t open on St. Patrick’s.
Billy Wilder (b. 1906)