Question

Let \(X\) and \(Y\) be compact Hausdorff spaces. The algebra generated by functions of the form \(f(x, y)=g(x) h(y)\), where \(g \in C(X)\) and \(h \in C(Y)\), is dense in \(C(X \times Y)\).

Short answer

The algebra is dense in \(C(X \times Y)\) due to the Stone-Weierstrass theorem.

Step-by-step solution

Understanding the Problem

Let us first understand what needs to be shown: Given compact Hausdorff spaces \(X\) and \(Y\), we have functions of the form \( f(x, y) = g(x)h(y) \) with \(g \in C(X)\) and \(h \in C(Y)\). We need to show that any function in \(C(X \times Y)\) can be approximated arbitrarily closely by finite linear combinations of functions like \(f(x, y)\), meaning they form a dense subset.

Use of Stone-Weierstrass Theorem

The Stone-Weierstrass theorem states that if \( A \) is a subalgebra of \( C(K) \), where \( K \) is a compact Hausdorff space, and \( A \) separates points and contains a non-zero constant function, then \( A \) is dense in \( C(K) \). Here, \( K = X \times Y \) and we want to show that the algebra generated by products of functions from \( C(X) \) and \( C(Y) \) satisfies the conditions of the Stone-Weierstrass theorem.

Subalgebra Condition

The set of functions \( f(x, y) = g(x)h(y) \) is closed under addition and multiplication. If \( g_1(x)h_1(y) \) and \( g_2(x)h_2(y) \) are two functions of this form, then:\[ (g_1g_2)(x) = g_1(x)g_2(x) \] and \[ (h_1h_2)(y) = h_1(y)h_2(y) \] ensure \( f_1 \cdot f_2 = (g_1g_2)(x)(h_1h_2)(y) \) remains in their span, satisfying the subalgebra condition.

Separating Points in \(X \times Y\)

To verify that the set separates points, take any two distinct points \((x_1, y_1), (x_2, y_2) \in X \times Y\). At least one of these coordinates is different, say \(x_1 eq x_2\), then there exists a continuous function \(g \in C(X)\) such that \(g(x_1) eq g(x_2)\). Set \(h(y) = 1\), resulting in \( f(x, y) = g(x)h(y)\) isolating the difference. Similarly, handle differences in \( y \).

Containment of Constant Functions

The constant function 1 can be expressed as \( f(x, y) = 1 imes 1 \), where each '1' belongs to their respective continuous function spaces, thus satisfying the requirement of containing a non-zero constant function.

Conclusion Using Stone-Weierstrass Theorem

Since the subalgebra: 1) Contains a non-zero constant function, 2) Separates points on \(X \times Y\), and 3) Is closed under multiplication, the Stone-Weierstrass theorem assures us that such an algebra is dense in \(C(X \times Y)\). This proves the original statement.

Key concepts

Compact Hausdorff Spaces

A compact Hausdorff space is a very prominent concept in topology. To understand this, imagine a space that is both neat and orderly, like a perfectly tidied room. Being "compact" means that the space is limited and well-contained, much like a room with a closed door. Mathematically, any collection of open sets that covers the space must have a finite subcollection that also covers the space. This property gives these spaces desirable traits, like every sequence having a convergent subsequence.
On the other hand, "Hausdorff" is a condition that ensures points can be distinguished from one another. Here, any two distinct points in the space can be contained in two non-overlapping open sets. This can be likened to assigning each point its personal territory.
Together, compactness and the Hausdorff property lay a solid foundation that makes analyzing functions on these spaces more manageable and predictable.

Dense Sets

A dense set is a fascinating concept in topology. Think of it like spices in a dish. Even though they're scattered, their influence reaches everywhere, providing flavor to every bite. In a mathematical sense, a set is dense if, no matter where you are in the space, you can find points of the dense set arbitrarily close to you. Essentially, you can approach any point in the entire space as closely as desired with points from the dense set.
In the context of the original problem, the algebra created by functions of form \(f(x, y) = g(x)h(y)\) is dense in the space of continuous functions over \(X \times Y\). This means any continuous function can be closely approximated by these specific types of functions, meeting consistency in flavor and structure.

Continuous Functions

Continuous functions maintain a smooth and predictable behavior, much like a story with no sudden jumps. These functions ensure there are no abrupt changes or leaps in their values as you move through the domain. Mathematical continuity means that small changes in the input result in small changes in the output.
In topological terms, a function \(f\) between two spaces is continuous if the pre-image of every open set is open. This feature is pivotal, as it allows us to connect different spaces in a consistent way, essential for proving complex theorems like the Stone-Weierstrass theorem.
In the original problem, we deal with continuous functions from compact Hausdorff spaces \(X\) and \(Y\), showing that such functions can approximate any desired output in \(C(X \times Y)\).

Algebraic Structures

Algebraic structures, at their core, provide a framework for understanding operations like addition and multiplication, just as grammar helps us form sentences. In mathematical spaces, an algebra involves a set equipped with operations that combine elements to produce another element in the set, following certain rules.
In this exercise, we consider an algebra generated by functions of the form \(f(x, y) = g(x)h(y)\). This is essentially a collection of organized operations. The product of these functions reflects a structured interplay of the continuous functions on \(X\) and \(Y\).
By proving that this algebra obeys the rules necessary for the Stone-Weierstrass theorem, we see that these structured combinations can densely fill the space of continuous functions on \(X \times Y\), illustrating the power of algebraic thinking in solving topological problems.