Problem 21 - Analysis Now

Question

(Contractible spaces.) A topological space \((X, \tau)\) is contractible if it is homotopic to a point (E 1.4.20). Show that \(X\) is contractible iff the identity map \({ }_{X}\) is homotopic to a constant map. Show that every convex subset of \(\mathbb{R}^{n}\) is contractible.

Step-by-Step Solution

Answer

A space is contractible if the identity map is homotopic to a constant map. Convex subsets of \(\mathbb{R}^n\) are contractible.

Step 1: Understanding Contractible Spaces

A topological space \((X, \tau)\) is said to be contractible if there is a homotopy from the identity map \(\text{id}_X\) on \(X\) to a constant map, which means \(X\) can be continuously deformed into a single point.

Step 2: Definition of Homotopy

Two maps \(f, g: X \to Y\) are homotopic, denoted by \(f \simeq g\), if there exists a continuous map \(H: X \times [0, 1] \to Y\) such that \(H(x, 0) = f(x)\) and \(H(x, 1) = g(x)\) for all \(x \in X\).

Step 3: Showing Condition of Contractibility

To show \(X\) is contractible if and only if the identity map is homotopic to a constant map, let \(H: X \times [0,1] \to X\) be the homotopy such that \(H(x, 0) = x\) (identity map) and \(H(x, 1) = c\) (constant map where \(c\) is a point in \(X\)). This demonstrates that \(X\) can indeed be continuously shrunk to a point.

Step 4: Contractibility of Convex Subsets

A subset \(A\) of \(\mathbb{R}^n\) is convex if for any two points \(x, y \in A\), the line segment connecting \(x\) and \(y\) lies entirely in \(A\). The map \(H(x, t) = (1-t)x + ty_0\) where \(y_0\) is a fixed point in \(A\), provides a homotopy from each point \(x\) to \(y_0\). Thus each \(x\) in \(A\) can be contracted to \(y_0\), showing that \(A\) is contractible.

Key Concepts

Homotopy Theory

Homotopy Theory is a fascinating part of algebraic topology that studies the idea of deforming shapes through continuous transformations. In the context of topological spaces, two maps, say \(f\) and \(g\), from a space \(X\) to a space \(Y\) are considered homotopic if there is a continuous deformation from one to the other. This notion is formalized by a continuous map \(H: X \times [0, 1] \rightarrow Y\), called a homotopy.

• At \(t = 0\), the map \(H(x, 0)\) is equivalent to \(f(x)\).
• At \(t = 1\), the map \(H(x, 1)\) is equivalent to \(g(x)\).

This gradual transformation from one map to another helps us understand when two spaces are essentially the same from a topological viewpoint. Homotopy theory provides a robust framework for investigating how shapes and spaces can morph into each other without tearing or gluing.

Convex Sets in Topology

In topology, a convex set is a specific type of subset within the Euclidean space \(\mathbb{R}^n\). A subset \(A\) is considered convex if, for any two points \(x\) and \(y\) within this set, the entire line segment connecting these points lies within the set. This simple definition has powerful implications in mathematics because it ensures a certain type of consistency and simplicity in the structure of the set.

• Convex sets are foundational in proving that such subsets are contractible.
• Any point in a convex set can be continuously "dragged" or contracted to another point within the same set.

The property of contractibility of convex sets often simplifies complex topological problems by providing pathways to relate them to simpler properties, such as the single point's topology.

Continuous Deformation

Continuous Deformation plays a central role in topology and homotopy theory. It implies an uninterrupted transformation, in which a shape or space is smoothly modified over a parameter \(t \in [0, 1]\). It conceptually resembles molding clay, where structures are altered without any breaks or gluing.

• Continuous deformation between two objects signifies that they are the same in the eyes of a topologist.
• This consistency is vital when determining if a space is equivalent to another, like deforming a space to a point.

This process underlies the formal definition of contractible spaces, adding depth to understanding their fundamental properties.

Topological Spaces

A topological space is one of the fundamental concepts in topology. It is a set \(X\) equipped with a topology \(\tau\), which is a collection of open sets that satisfy particular properties:

• The entire set \(X\) and the empty set are included in \(\tau\).
• The intersection of finitely many open sets is also an open set.
• The union of any collection of open sets is an open set.

Topological spaces allow us to generalize the notion of geometric shapes and explore their properties without relying on specific metrics. They pave the way for expressing complex ideas such as continuity, convergence, and connectedness, which are all interpreted through the lens of topology.

Identity Map Homotopy

Identity Map Homotopy is a special concept in topology used to demonstrate when a space can be continuously deformed into a single constant shape or point. Specifically, it describes a homotopy that relates the identity map—where every point maps to itself—to a constant map, where every point maps to a single point \(c\) in the space.

• This homotopy \(H: X \times [0, 1] \rightarrow X\) must satisfy: \(H(x, 0) = x\) and \(H(x, 1) = c\).
• Such a transformation indicates that the space is contractible.

Identity map homotopy strongly indicates the flexibility and deformability of a space, being a crucial piece in understanding contractible spaces.