Problem 14 - A First Course in Algebraic Topology

Question

Let \(X\) be a compact Hausdorff space and let \(U\) be an open subset of \(\mathrm{X}\) not equal to \(\mathrm{X}\) itself. Prove that $$ \mathrm{U}^{\infty} \cong \mathrm{X} /(\mathrm{X}-\mathrm{U}) $$ (Hint: Consider h: \(U^{\infty} \rightarrow X /(X-U)\) given by \(h(u)=p(u)\) for \(u \in U\) and \(\mathrm{h}(\infty)=\mathrm{p}(\mathrm{X}-\mathrm{U})\) where \(\mathrm{p}: \mathrm{X} \rightarrow \mathrm{X} /(\mathrm{X}-\mathrm{U})\) is the natural projection.) Deduce that if \(x \in X\) (and \(X\) is a compact Hausdorff space) then $$ (\mathrm{X}-\\{\mathrm{x}\\})^{\infty} \cong \mathrm{X} $$.

Step-by-Step Solution

Answer

\( U^{\infty} \) is homeomorphic to \( X/(X-U) \) by identifying the added point with the quotient point, and \((X - \{x\})^{\infty} \cong X\).

Step 1: Define the spaces involved

Let's first understand what a one-point compactification is. The space \( U^{\infty} \) is the one-point compactification of \( U \). Given that \( U \) is an open subset of \( X \), \( X - U \) is a closed subset of \( X \). The notation \( X / (X-U) \) refers to the quotient space where all points in \( X - U \) are identified to a single point.

Step 2: Define the map h

Define the map \( h: U^{\infty} \rightarrow X / (X-U) \) by \( h(u) = p(u) \) for \( u \in U \) and \( h(\infty) = p(X-U) \). The map \( p: X \rightarrow X / (X-U) \) is the natural projection that sends every point in \( X-U \) to the single equivalence class \( [X-U] \).

Step 3: Show h is continuous

To show that \( h \) is continuous, note that the restriction \( h|_U \) is the composition of continuous functions \( p|_U: U \rightarrow X / (X-U) \). For the point at infinity, \( h(\infty) = p(X-U) \) corresponds to closing up the space as \( X-U \) is identified to a point, satisfying continuity in one-point compactifications.

Step 4: Show h is a homeomorphism

To prove \( h \) is a homeomorphism, we demonstrate a continuous inverse. Define \( g: X / (X-U) \rightarrow U^{\infty} \) by mapping the equivalence class \( [X-U] \) to \( \infty \) and each \([u] \) for \( u \in U \) back to \( u \). Both \( h \) and \( g \) are continuous and inverses of each other, hence \( h \) is a homeomorphism.

Step 5: Apply result to specific point set

Now, let's deduce the given result. For the case \( X - \{x\} \), consider \( U = X - \{x\} \). The space \( X \) with a point removed is an open subset. By compactifying \( X - \{x\} \) at infinity, we regain \( X \) by effectively adding back the removed point as the point \( \infty \). Thus, \( (X - \{x\})^{\infty} \cong X \).

Key Concepts

Compact Hausdorff Space

A compact Hausdorff space is a type of topological space that possesses two very valuable properties: compactness and the Hausdorff condition. Let's explore what these terms mean separately and then together.

• Compactness: A space is compact if every open cover has a finite subcover. This basically means that the space can be "covered" by a finite number of open sets, regardless of how many you start with. This is equivalent to the fact that a space is compact if every sequence has a convergent subsequence within the space.


• Hausdorff Condition: A space is Hausdorff (also known as being "T2") if any two distinct points can be separated into distinct open sets. This ensures that points are "well-behaved" and not crowding each other, which allows for better control over limits and continuity.

Together, a compact Hausdorff space becomes a very well-behaved setting in topology. They ensure that the space is both bounded in a certain sense and that limits behave nicely, which is why they are well-studied in algebraic topology.

Understanding these properties helps in analyzing how spaces like \( U^{\infty} \) behave when compactifying open subsets of compact Hausdorff spaces.

One-Point Compactification

One-point compactification of a topological space \( U \) involves adding a single "ideal" point, often denoted as \( \infty \), to make the space compact. Here’s a closer look at how this process works:

• Starting with a Non-Compact Space: Begin with a non-compact space like an open subset \( U \) of a larger space \( X \). Our goal is to transform \( U \) into a compact space.


• Adding the Point at Infinity: By adding \( \infty \), you create \( U^{\infty} \), compactifying \( U \) by making sure that all the "ends" of \( U \) are closed up by \( \infty \).


• Topological Alterations: The point at infinity needs to adhere to the topology of \( U \). Thus, a set is open in \( U^{\infty} \) if it is open in \( U \) or if it includes \( \infty \) and its complement in \( U \) is compact.

The one-point compactification is useful because it addresses non-compact spaces, transforming them into compact ones, which are easier to manage within the realms of algebraic topology and further aligns with the properties observed in compact Hausdorff spaces.

Quotient Space

Quotient spaces are formed by taking an existing space and identifying points, creating equivalence classes that simplify or change the structure of the space.

• Creating Equivalence Classes: In the context of the exercise, consider \( X / (X-U) \). Here, all points in \( X - U \) are identified as one single point. This single point represents the equivalence class for all points part of or equivalent to \( X-U \).


• Simplifying the Space: By identifying equivalent points, we can simplify complex or difficult spaces, making topological properties easier to study and understand.


• Projection Maps: The natural projection map \( p: X \rightarrow X / (X-U) \) is central, consolidating points into their respective classes, which forms the basis for defining other functions or maps in proofs, such as the one in the exercise.

Understanding quotient spaces helps in analyzing how spaces transform under certain identifications, allowing algebraic topology to tackle problems from a simplified perspective. For instance, understanding \( X/(X-U) \) facilitates proving homeomorphisms.

Homeomorphism

Homeomorphism is a key concept in topology indicating when two spaces are essentially "the same" in a topological sense, even if they appear different geometrically.

• Definition: A function \( h: A \rightarrow B \) is a homeomorphism if it is a bijective, continuous function whose inverse \( h^{-1} \) is also continuous.
• Structure Preservation: Homeomorphisms preserve the structure of spaces, meaning properties like connectedness and compactness remain unchanged between the spaces \( A \) and \( B \).
• Proving Homeomorphisms: In the exercise, we demonstrate a homeomorphism between \( U^{\infty} \) and \( X/(X-U) \) by finding continuous functions \( h \) and \( g \) (as its inverse), confirming the spaces' equivalence in a topological sense.

Recognizing when two spaces are homeomorphic enables topologists to conclude that they share fundamental properties, providing a powerful tool for classifying spaces within algebraic topology. The concept simplifies complex analysis, showcasing equivalences that would otherwise be hidden.