Irreducible Components: Definition, Properties and Applications

Jérémy Celse^*
*Correspondence: Jérémy Celse, Department of Management, University of Salford, The Crescent, Salford, Manchester, M5 4WT, UK, Email:

Keywords

Diagnostic and Statistical Manual of Mental Disorders (DSM) • International Classification of Diseases (ICD) • Research Domain Criteria (RD°C)

Introduction

Let X be a topological space. A subset Y ⊆ X is said to be irreducible if it satisfies the following two conditions: Y is non-empty and closed. If Y can be expressed as the union of two proper closed subsets Y1 and Y2, then one of Y1 or Y2 must be equal to the empty set. In other words, an irreducible subset is one that cannot be divided into two proper closed subsets without one of them being empty. The maximal irreducible subsets of X are called its irreducible components.

Literature Review

Let's look at some examples to help illustrate the concept of irreducible components.

Consider the real line R with its usual topology. Then any interval [a, b] with a < b is an irreducible subset of R. To see why, suppose [a, b] can be expressed as the union of two proper closed subsets [a, c] and [c, b] with a < c < b. Then either a ≤ c < b or a < c ≤ b. Without loss of generality, assume the former. Then c is a limit point of [a, c], so [a, c] is not closed. Therefore, [a, c] is not a proper closed subset of [a, b], and we have a contradiction [1].

Consider the plane R2 with its usual topology. Then any line through the origin is an irreducible subset of R2. To see why, suppose L is a line through the origin that can be expressed as the union of two proper closed subsets L1 and L2. Then there exists a point p in L1 that is not on L2. Without loss of generality, assume that p lies above L2. Then the segment of L connecting p to the origin is a closed subset of L that does not contain any points of L2. Therefore, L1 contains this segment, and L2 does not, contradicting the assumption that L1 and L2 are both proper closed subsets of L [2].

Consider the space Pn of polynomials of degree at most n with complex coefficients, with the usual topology. Then the irreducible subsets of Pn are precisely the zero sets of irreducible polynomials in C[x1, ..., xn+1] of degree at most n. To see why, suppose Y is an irreducible subset of Pn. Then there exists an irreducible polynomial f in C[x1, ..., xn+1] of degree at most n such that Y is the zero set of f in Pn. Conversely, if f is an irreducible polynomial in C[x1, ..., xn+1] of degree at most n, then the zero set of f in Pn is an irreducible subset of Pn [3-6].

Discussion

There are several important properties of irreducible components that we will discuss below.The irreducible components of a space are maximal in the sense that they cannot be enlarged without ceasing to be irreducible. In other words, if Y is an irreducible subset of X and Z is a closed subset of X such that Y ⊆.

Irreducible components are an important concept in mathematics, particularly in algebraic geometry and commutative algebra. In this article, we will define and explore irreducible components, including their properties, uses, and applications.

Definition of irreducible component

An irreducible component of a topological space is a nonempty closed subset that is maximal with respect to being irreducible. An irreducible set is one that cannot be written as the union of two proper closed subsets. Therefore, an irreducible component of a topological space is a closed subset that cannot be written as the union of two proper closed subsets that are also closed subsets of the space.

Properties of irreducible components

Irreducible components have several properties that make them important in mathematics. One such property is that they are maximal with respect to being irreducible. This means that any subset of an irreducible component that is also irreducible must be the irreducible component itself. Additionally, any two irreducible components of a topological space are either equal or disjoint.

Another property of irreducible components is that they are closed subsets of the topological space. This follows directly from the definition, as an irreducible component is a nonempty closed subset that is maximal with respect to being irreducible. Additionally, every closed subset of a topological space can be written as the union of finitely many irreducible components.

Applications of irreducible components

Irreducible components have several important applications in mathematics. One such application is in algebraic geometry. In this field, a variety is a geometric object that is defined by a system of polynomial equations. The irreducible components of a variety correspond to the connected components of the set of solutions to the polynomial equations.

For example, consider the variety defined by the equation x^2 + y^2 = 1 in the complex plane. This variety consists of two irreducible components: the unit circle and the single point (-1,0). The unit circle is irreducible because it cannot be written as the union of two proper closed subsets. The single point (-1,0) is also irreducible because it is a single point.

Another application of irreducible components is in commutative algebra. In this field, an ideal is a subset of a ring that satisfies certain properties. The irreducible components of an ideal correspond to the prime ideals that contain the original ideal.

For example, consider the ideal (x^2,xy) in the polynomial ring k[x,y], where k is a field. The irreducible components of this ideal correspond to the prime ideals that contain the ideal (x^2,xy). It can be shown that there are two prime ideals that contain this ideal: (x) and (x,y). Therefore, the irreducible components of the ideal (x^2,xy) are (x) and (x,y).

Irreducible components are closely related to the notion of connectedness. In fact, a topological space is connected if and only if it has exactly one irreducible component. This follows from the fact that any two irreducible components of a topological space are either equal or disjoint. Therefore, if a topological space has more than one irreducible component, it cannot be connected.

For example, consider the topological space consisting of the union of the unit circle and the single point (-1,0) in the complex plane. This space has two irreducible components, the unit circle and the single point (-1,0). Therefore, it is not connected.

Conclusion

Irreducible components are also related to the dimension of a topological space. In fact, the dimension of a topological space is equal to the maximum of the dimensions of its irreducible components.

Acknowledgement

None.

Acknowledgement

None.

Conflict of Interest

No conflict of interest.

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