91Ó°ÊÓ

(Homotopy.) Two continuous maps \(f: X \rightarrow Y\) and \(g: X \rightarrow Y\) between topological spaces \(X\) and \(Y\) are homotopic if there is a continuous function \(F:[0,1] \times X \rightarrow Y\) (where \([0,1] \times X\) has the product topology), such that \(F(0, x)=f(x)\) and \(F(1, x)=g(x)\) for every \(x\) in \(X\). Intuitively speaking, the homotopy \(F\) represents a continuous deformation of \(f\) into \(g\). Show that any continuous function \(f: \mathbb{R}^{n} \rightarrow Y\) is homotopic to a constant function, and that the same is true for any continuous function \(g: X \rightarrow \mathbb{R}^{n}\). Show that the identity function r. \(\mathrm{S}^{1} \rightarrow \mathrm{S}^{1}\) [where \(\mathrm{S}^{1}=\left\\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\\}\) ] is not homotopic to a constant function.

Step by step solution

01

Setting Up the Homotopy for Functions from \( \mathbb{R}^n \)

To show any continuous function \( f: \mathbb{R}^n \rightarrow Y \) is homotopic to a constant function, consider the homotopy \( F: [0,1] \times \mathbb{R}^n \rightarrow Y \) defined by \( F(t, x) = f(tx) \). When \( t = 0 \), \( F(0, x) = f(0) \), which is constant. When \( t = 1 \), \( F(1, x) = f(x) \), recovering the original function, thus \( f \) is homotopic to the constant function \( f(0) \).

02

Homotopy for Functions into \( \mathbb{R}^n \)

To show any continuous function \( g: X \rightarrow \mathbb{R}^n \) is homotopic to a constant function, use the homotopy \( G: [0,1] \times X \rightarrow \mathbb{R}^n \) defined by \( G(t, x) = (1-t)g(x) + tc \) for some constant point \( c \in \mathbb{R}^n \). At \( t=0 \), \( G(0, x) = g(x) \); at \( t=1 \), \( G(1, x) = c \), hence \( g \) is homotopic to the constant function \( c \).

03

Identity Function on \( S^1 \) is not Homotopic to a Constant Function

The identity function \( ext{id}: S^1 \rightarrow S^1 \) cannot be contracted to a point continuously because \( S^1 \) is not contractible. Suppose such a homotopy \( H: [0,1] \times S^1 \rightarrow S^1 \) exists with \( H(0, x) = ext{id}(x) \) and \( H(1, x) = c \) for some constant \( c \). The existence of such a homotopy would imply that \( S^1 \) is homotopically equivalent to a point, contradicting the fundamental group property \( \pi_1(S^1) eq 0 \). Therefore, it is not possible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Topological Spaces

In the world of mathematics, topological spaces are an abstraction of geometric concepts. A topological space consists of a set along with a collection of open sets that define a structure known as a topology. Imagine these as flexible spaces where certain rules about "closeness" apply.
In simpler terms, consider how you can deconstruct shapes into sets of points. A topological space defines how these points relate through a topology, without worrying about the exact shape itself.
Important aspects include:

• Open Sets: These are the building blocks. They define the topology on a set.
• Continuous Functions: These respect the topology by sending close points to close points.

Topological spaces provide the framework to discuss concepts like continuity and convergence that are crucial for understanding deeper properties of spaces.

Continuous Maps

Continuous maps are essential in topology, describing functions that preserve the notion of closeness. They ensure maps do not tear or glue parts together in inappropriate ways.
A function between two topological spaces is continuous if, for every open set in the target space, the pre-image under the function is also open in the domain space.
Here’s why they are important:

• Preservation of Structure: They maintain the intrinsic topological structure of a space.
• Homotopy: Two functions are homotopic if one can be continuously deformed into the other by a continuous map over time.

Understanding them is akin to recognizing how familiar transformations like stretching or shrinking occur without breaking the object. Think of a clay model that can be reshaped continuously without tearing.

Identity Function

The identity function is a straightforward concept. For every input, it returns the input itself unchanged. In mathematical terms, for a space \( X \), the identity function \( \text{id}_X \) is defined by \( \text{id}_X(x) = x \) for every \( x \) in \( X \).
This function is a base case in many contexts, as it serves as a benchmark that retains the entire content of the domain exactly as it is.
In topological terms, the identity function is always continuous because pre-images of open sets remain open. It's often used in arguments and proofs that involve deformation, such as determining non-trivial transformations, because it shows that the original space is indispensable in certain mappings.

Constant Function

A constant function maps every input to a single value in the target space. No matter what the input is, the output stays the same.
Formally, for a function \( f: X \rightarrow Y \), it is constant if there exists a \( c \in Y \) such that \( f(x) = c \) for all \( x \) in \( X \).
Key features include:

• Simplicity: The function is easy to compute and analyze.
• Continuity: Always continuous as the pre-image of any set is either empty or the entire space.
• Homotopy with Other Functions: In many settings, complex functions can be continuously deformed into constant functions, showing flexibility and connectedness within the space.

Constant functions play an essential role in understanding how spaces can be contracted to a point, illustrating basic concepts of deformation and equivalence in topology.