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.

Read more about Closed Set: Equivalent Definitions of A Closed Set, Properties of Closed Sets, Examples of Closed Sets, More About Closed Sets

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

... An arbitrary intersection of open

**sets**in X is open ...

**Closed set**characterization ... An arbitrary union of

**closed sets**in X is

**closed**...

... 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 ...

... 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 ...

... 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 ...

**Closed Set**s

... 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)