/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Show that any vertex in a connec... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 13

Show that any vertex in a connected finite tree is a strong deformation retract of the tree.

Short Answer

Expert verified
A vertex in a connected finite tree is a strong deformation retract because we can construct a continuous homotopy shrinking the tree to that vertex.

Step by step solution

01

Understanding the Tree Structure

A tree is a connected acyclic graph. In a tree with `n` vertices, there are `n-1` edges, and removing any edge disconnects the tree into separate components.
02

Define a Retraction

A retraction is a continuous map (function) from a space to a subspace which behaves as the identity map on that subspace. In this case, we need a map from the tree `T` to a single vertex `v` such that each point on the tree can be continuously deformed to `v`.
03

Construct the Retraction Map

Consider a tree `T` with vertex `v`. Define a map `f: T 鈫� T` as follows: every vertex maps to itself, and every edge incident to `v` collapses to `v`, forming paths from any vertex to `v`. Express `T` as the union of paths from each vertex to `v`, mapping any path point by point to `v`.
04

Prove Continuity of the Map

For a tree, continuity of this retraction implies showing no jumps or breaks in the mapping occur. Since paths are linear and shrinking these paths into `v` straightforwardly maintains connectedness, we can ensure the retraction is continuous at each step, producing a strong deformation retraction.
05

Verify Strong Deformation Retraction

A strong deformation retraction requires not only a continuous map but also a continuous homotopy from the identity map on `T` to the retraction map `f`. Define a homotopy `H:T x [0,1] 鈫� T` by `H(x, t)` reduces the distance from any vertex `x` to `v` linearly with `t`. At `t=1`, every point resides at `v`, satisfying the strong deformation retraction conditions.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Tree Structure
Trees are a fundamental concept in mathematics and computer science. A tree is a special kind of graph that is both connected and acyclic, meaning it has no loops or closed paths. Here's a quick breakdown of its features:
  • The vertices of a tree are like points or nodes, while the edges are the lines that connect these points.
  • In any tree with `n` vertices, there are always `n-1` edges. This is crucial because adding even one more edge would create a cycle, breaking its acyclic nature.
  • Removing any edge from a tree splits it into two distinct components, which emphasizes its connected nature.
These properties allow trees to effectively represent hierarchical structures, such as organizational charts or family trees. Recognizing these characteristics helps in understanding how a vertex within this structure can be analyzed for particular mappings, as seen in this exercise.
Retraction Map
In topology, a retraction map is a way to 'shrink' a space onto a subspace without tearing or gluing pieces of the space. More formally:
  • A retraction is a continuous function from a space to a subspace that behaves like the identity map over the subspace.
  • Imagine a rubber sheet stretched and then smoothly deformed, bringing all parts of the sheet towards a specific central point or region.
In the context of a tree, our "rubber sheet" is deformed toward one vertex, mapping each point on the tree continuously to this single vertex. This transformation is essential in showing how each part of the tree can be pulled into one point while maintaining the integrity of the tree's connections.
Continuity
Continuity is a central concept in mathematics, especially in topology. A function is continuous if small changes in the input produce small changes in the output, with no sudden jumps or breaks. For our retraction map:
  • The function must smoothly map from each vertex or path back to a selected vertex `v`.
  • This is often visualized as a path; as you trace along the path, your hand moves smoothly, not jumping between disconnected points.
The idea here is to ensure that as we collapse or transform the tree during retraction, the path from any given point in the tree to the vertex `v` remains uninterrupted and connected. Continuous deformations preserve the essential connected nature of the structure, even while altering its form.
Strong Deformation Retraction
A strong deformation retraction is an enhanced version of a simple retraction. It doesn't just map the points of the space to the subspace; it continuously moves the entire space towards the subspace across a range of time steps:
  • Consider a "time" parameter `t` that varies from 0 to 1, mapping how every point in the space is pushed towards our designated vertex as time progresses.
  • At time `t=0`, each point sits at its original position, and by `t=1`, every point is drawn into the vertex `v`.
  • This requires defining a homotopy, a continuous deformation function `H(x,t)` that represents this movement through time.
By constructing this time-dependent path, we establish that the tree edges and vertices do not lose their connections or relationships during the transformation, demonstrating strong deformation retraction. This concept solidifies the connection between topology and geometry, describing transformations in a rigorously defined manner.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Given any collection \(\left\\{X_{\alpha}: \alpha \in A\right\\}\) of topological spaces, show that their disjoint union \(\mathrm{I}_{\alpha} X_{\alpha}\), together with the disjoint union topology and the natural inclusions \(\iota_{\alpha}: X_{\alpha} \hookrightarrow \mathrm{L}_{\alpha} X_{\alpha}\), is their sum in the category TOP.

Let \(B \subset \mathbb{R}^{n}\) be any convex set, \(X\) any topological space, and \(A\) any subset of \(X\). Show that any two continuous maps \(f, g: X \rightarrow B\) that agree on \(A\) are homotopic relative to \(A\).

Given any collection \(\left\\{X_{\alpha}: \alpha \in A\right\\}\) of topological spaces, define a basis in the Cartesian product \(\prod_{\alpha} X_{\alpha}\) consisting of product sets of the form \(\prod_{\alpha} U_{\alpha}\), where \(U_{\alpha}\) is open in \(X_{\alpha}\) and \(U_{\alpha}=X_{\alpha}\) for all but finitely many \(\alpha\). Show that this is a basis, and that \(\prod_{\alpha} X_{\alpha}\) with this topology is the product of the spaces \(X_{\alpha}\) in the category TOP.

For any compact surface \(S\), show that \(S\) with one point removed is homotopy equivalent to a bouquet of finitely many circles.

Let \(X\) be a path connected space and \(p, q \in X\). Determine an algebraic necessary and sufficient condition on the fundamental group of \(X\) under which all path classes from \(p\) to \(q\) give the same isomorphism of \(\pi_{1}(X, p)\) with \(\pi_{1}(X, q)\).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.