Problem 16 - A First Course in Algebraic Topology

Question

Suppose that \(X\) is a compact Hausdorff space and that \(A\) is a closed subspace of \(\mathrm{X}\). Suppose furthermore that \(\mathrm{A}\) is a \(\mathrm{G}\)-space with \(\mathrm{G}\) finite. Define a relation \(\sim\) on \(X\) by saying that \(x \sim x^{\prime}\) if and only if either \(x=x^{\prime}\) or both \(x, x^{\prime} \in A\) and \(x=g \cdot x^{\prime}\) for some \(g \in G\). Prove that \(\sim\) is an equivalence relation on \(X\) and prove that the space \(\mathrm{X} / \sim\) is Hausdorff.

Step-by-Step Solution

Answer

\(\sim\) is an equivalence relation, and \(X/\sim\) is Hausdorff.

Step 1: Understand the Set-Up

We are given a compact Hausdorff space \( X \) and a closed subspace \( A \) that is also a \( G \)-space, with \( G \) being a finite group. We have to define an equivalence relation \( \sim \) on \( X \) based on these conditions.

Step 2: Define the Equivalence Relation \(\sim\)

The relation \( x \sim x' \) is defined in two cases: (1) \( x = x' \) or (2) both \( x, x' \in A \) and there exists a \( g \in G \) such that \( x = g \cdot x' \). This definition models the orbit relation of \( G \) on \( A \), identifying points in \( A \) related by group elements.

Step 3: Prove Reflexivity of \(\sim\)

For any \( x \in X \), clearly \( x = x \). Furthermore, for any \( x \in A \), \( x = e \cdot x \), where \( e \) is the identity element of the group \( G \). Hence, the relation is reflexive because \( x \sim x \) for all \( x \in X \).

Step 4: Prove Symmetry of \(\sim\)

Assume \( x \sim x' \). If \( x = x' \), then obviously \( x' = x \). If \( x, x' \in A \) and \( x = g \cdot x' \) for some \( g \in G \), then \( x' = g^{-1} \cdot x \) and since \( G \) is a group, \( g^{-1} \) is also in \( G \), hence \( x' \sim x \). Thus, \( \sim \) is symmetric.

Step 5: Prove Transitivity of \(\sim\)

Assume \( x \sim x' \) and \( x' \sim x'' \). If \( x = x' \) or \( x' = x'' \), then \( x \sim x'' \) holds trivially. If \( x, x', x'' \in A \), there exist \( g, h \in G \) such that \( x = g \cdot x' \) and \( x' = h \cdot x'' \). Therefore, \( x = (g \cdot h) \cdot x'' \), and since \( g \cdot h \in G \), we get \( x \sim x'' \). Thus, \( \sim \) is transitive.

Step 6: Conclude that \(\sim\) is an Equivalence Relation

Since \( \sim \) is reflexive, symmetric, and transitive, it is an equivalence relation on \( X \).

Step 7: Show \( X/\sim \) is a Hausdorff Space

The quotient space \( X/\sim \) can be viewed as identifying all elements in \( A \) that are equivalent under the group action. Since \( X \) is compact and Hausdorff, we only need to show distinct equivalence classes have disjoint neighborhoods. For \( x ot\sim y \), if \( x, y \in X \setminus A \), use Hausdorffness of \( X \). If \( x, y \in A \) but distinct orbits, use closed sets and continuity of group action to separate them, relying on finiteness of \( G \). If one in \( A \) and one not, separate using closedness of \( A \) in \( X \). Thus, \( X/\sim \) is Hausdorff.

Key Concepts

Compact Hausdorff Space

A Compact Hausdorff Space is a type of topological space with certain properties that make it particularly nice to work with in the field of mathematics, especially in algebraic topology. This type of space involves two main characteristics:

• Compactness: A space is compact if every open cover of the space has a finite subcover. This means that you can take any collection of open sets that completely cover the space and there will be some finite subset of those sets that also cover the entire space.
• Hausdorff Condition: A space is Hausdorff (or separated) if for any two distinct points, there exist neighborhoods around each point that do not overlap. This ensures that points can be "separated" by open sets.

Compact Hausdorff spaces are important because they enjoy many pleasant properties. For example, they are always normal spaces, which means any two disjoint closed sets can be separated by disjoint open sets. Additionally, every continuous image of a compact space is compact, and every closed subset of a compact space is compact itself. These properties make such spaces useful in various mathematical disciplines.

Equivalence Relation

An Equivalence Relation is a way of grouping or classifying elements in a set according to certain criteria, allowing us to discuss concepts of sameness or equality in a broader sense than just being exactly the same object. An equivalence relation must satisfy three properties:

• Reflexivity: Every element is equivalent to itself, meaning for any element \( x \), \( x \sim x \).
• Symmetry: If one element is equivalent to another, then the second element is equivalent to the first. For instance, if \( x \sim y \), then \( y \sim x \).
• Transitivity: If an element is equivalent to a second, and the second to a third, then the first is equivalent to the third. For elements \( x, y, \) and \( z \), this means if \( x \sim y \) and \( y \sim z \), then \( x \sim z \).

In the context of the exercise, the equivalence relation defined on the space \( X \) serves to group elements that are either identical or are related through a symmetry under a group action, reflecting a form of symmetry inherent in the system.

Quotient Space

A Quotient Space is formed by taking a topological space and an equivalence relation on it, then considering the set of equivalence classes as a new topological space. In simpler terms, a quotient space "glues together" points that are equivalent under the defined relation.

• The equivalence classes under the equivalence relation become the points of this new space. For example, if you have a line and an equivalence relation that identifies every point with its mirror image, the quotient space might be a circle.
• The topology of the quotient space is defined such that a set is open if and only if its pre-image under the natural projection is open in the original space.

In this exercise, \( X/\sim \) represents the quotient space by the equivalence relation \( \sim \). This new space identifies points in \( A \) based on the group action given by \( G \), effectively "factoring out" the symmetry due to \( G \). The challenge is to show this space retains Hausdorffness, ensuring it's a well-behaved topological structure.

Group Action

A Group Action on a space provides a structured way for a group to "act on" or transform elements within the space. It is a powerful tool for studying symmetries and transformations. A group action is defined by two main criteria:

• Identity Action: The identity element of the group acts trivially on any element of the set, meaning it does nothing to it. If \( e \) is the identity element and \( x \) is an element in the space, then \( e \cdot x = x \).
• Compatibility: Group action respects the group operation, meaning for any two group elements \( g \) and \( h \), and any element \( x \) in the space, \( g \cdot (h \cdot x) = (g \cdot h) \cdot x \).

In the exercise, \( G \) acts on the closed subspace \( A \). This means each "symmetry" in \( G \) applies to \( A \), helping to establish the equivalence relation by comparing which elements can be transformed into one another by the group's actions. This setup is crucial in ensuring that the resulting quotient space is well-defined and inherits certain properties like compactness.