Search results
Results from the WOW.Com Content Network
A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [5] A set can never been considered as open by itself.
Since any set is open, the complement of any set is open too, and therefore any set is closed. So, all sets in this metric space are clopen. As a less trivial example, consider the space Q {\displaystyle \mathbb {Q} } of all rational numbers with their ordinary topology, and the set A {\displaystyle A} of all positive rational numbers whose ...
The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense. Singleton points (and thus finite sets) are closed in T 1 spaces and Hausdorff spaces. The set of integers is an infinite and unbounded closed set in the real numbers.
These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following. In any discrete space, since every set is closed (and also open), every set is equal to its closure.
Relatively compact set; Regular open set, regular closed set; Connected set; Perfect set; Meagre set; Nowhere dense set; Relative to a metric. Bounded set; Totally ...
The closed interval [,) in the standard subspace topology is connected; although it can, for example, be written as the union of [,) and [,), the second set is not open in the chosen topology of [,). The union of [ 0 , 1 ) {\displaystyle [0,1)} and ( 1 , 2 ] {\displaystyle (1,2]} is disconnected; both of these intervals are open in the standard ...
The closed sets are the unions of finitely many pairs , +, or the whole set. The open sets are the complements of the closed sets; namely, each open set consists of all but a finite number of pairs 2 n , 2 n + 1 , {\displaystyle 2n,2n+1,} or is the empty set.
This is equivalent to the condition that the preimages of the open (closed) sets in Y are open (closed) in X. In metric spaces, this definition is equivalent to the ε–δ-definition that is often used in analysis. An extreme example: if a set X is given the discrete topology, all functions :