How do you find the closure of a topological space?

Let (X,τ) be a topological space and A be a subset of X, then the closure of A is denoted by ¯A or cl(A) is the intersection of all closed sets containing A or all closed super sets of A; i.e. the smallest closed set containing A.

What is the discrete topological space?

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set.

What are all open and closed sets in discrete metric space?

Now for every subset A of X, Ac = X\A is a subset of X and thus Ac is a open set in X. This implies that A is a closed set. Thus every subset in a discrete metric space is closed as well as open. For any point x ∈ X, {x} is a open set containing no other point of X, in particular it can not serve as a limit point of A.

Is a discrete set open or closed?

A set is discrete if it has the discrete topology, that is, if every subset is open. …

What is the closure of a space?

In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S.

What is closure set in topological space?

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.

What is the closure of a set?

What sets are closed in the discrete topology?

Since it is a discrete topology the compliment to each subset must be in the topology and therefore every set is closed. By definition if the topology has every set of X it is a discrete topology. Since it is a discrete topology the compliment to each subset must be in the topology and therefore every set is closed.

What is open and close set?

Properties. The union of any number of open sets, or infinitely many open sets, is open. The intersection of a finite number of open sets is open. A complement of an open set (relative to the space that the topology is defined on) is called a closed set.

What is set in discrete?

A set is an unordered collection of different elements. A set can be written explicitly by listing its elements using set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.

How do you show a topology is discrete?

Discrete Topology is Topology

1. Let S be a set.
2. Let τ be the discrete topology on S.
3. Let T=(S,τ) be the discrete space on S.
4. Then by definition τ=P(S), that is, is the power set of S.
5. We confirm the criteria for T to be a topology:
6. ◼

What is closed and closure?

By idempotency, an object is closed if and only if it is the closure of some object. These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set. If X is contained in a set closed under the operation then every subset of X has a closure.

How do you find the topology of a closed set?

Thus the set τ of all closed sets in the interval [a, b] provide a topology for X = [a, b]. The set X = [a, b] with the topology τ represents a topological space. In Fig. 2 is depicted a typical open set, closed set and general set in the plane where dashed lines indicate missing boundaries for the indicated regions.

What is the topological space of a set?

Topological space (X, τ). A topological space X with topology τ is often referred to as the topological space (X, τ). The collection τ of open sets defining a topology on X doesn’t represent all possible sets that can be formed on X. Let π be the set of all possible sets that can be formed on X.

Does the closure of a set depend on the underlying space?

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.

What is the topology of X?

A topology on a set X then consists of any collection τ of subsets of X that forms a closed system with respect to the operations of union and intersection. Notation. Topological space (X, τ). A topological space X with topology τ is often referred to as the topological space (X, τ).