New topologies from old via ideals

A fascinating topic!

In mathematics, ideals are a fundamental concept in abstract algebra, particularly in ring theory. An ideal is a subset of a ring that is closed under addition and multiplication by elements of the ring, and also contains the product of any element of the ideal and any element of the ring.

Now, let's explore how ideals can be used to create new topologies from old ones.

Ideal-based constructions

Given a topological space X and an ideal I in the ring of continuous functions C(X), we can construct a new topological space X/I as follows:

1. Define a relation on X by saying that two points x, y are equivalent if and only if there exists a function f in I such that f(x) = f(y).
2. The equivalence classes of this relation are called the "cosets" of I, denoted by [x]_I.
3. The set of all cosets [x]_I, denoted by X/I, becomes a topological space by defining the topology as the quotient topology, where the open sets are the sets of the form [U]_I = { [x]_I | x ∈ U }, where U is an open set in X.

Properties of X/I

The space X/I inherits some properties from X:

• The space X/I is Hausdorff if and only if X is Hausdorff and I is a proper ideal (i.e., I ≠ C(X)).
• The space X/I is compact if and only if X is compact and I is a finitely generated ideal.
• The space X/I is connected if and only if X is connected and I is a radical ideal (i.e., I is the intersection of all prime ideals containing it).

Examples

1. Sierpinski space: Let X be the real line ℝ, and I be the ideal of all continuous functions that vanish at infinity. Then X/I is the Sierpinski space, which is a compact, Hausdorff space with two points.
2. Tychonoff plank: Let X be the unit interval [0, 1], and I be the ideal of all continuous functions that are constant on some non-empty open set. Then X/I is the Tychonoff plank, which is a compact, Hausdorff space with two points.

Applications

Ideal-based constructions have applications in various areas of mathematics and computer science, such as:

• Topology: Ideal-based constructions can be used to create new topological spaces with interesting properties, such as compactness, connectedness, or Hausdorffness.
• Algebraic geometry: Ideal-based constructions can be used to study the geometry of algebraic varieties, such as the study of singularities and deformations.
• Computer science: Ideal-based constructions can be used in computer science to study the complexity of algorithms and the structure of data structures.

In conclusion, ideals are a powerful tool for creating new topologies from old ones, and their applications are diverse and fascinating.