/*! 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 5 Let \(f\) and \(g\) be continuou... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 5

Let \(f\) and \(g\) be continuous functions between topological spaces \(X\) and \(Y\), where \(Y\) is a Hausdorff space. Show that the set $$ \\{x \in X \mid f(x)=g(x)\\} $$ is closed in \(X\). Show that if \(f\) and \(g\) are equal on a dense subset of \(X\), then \(f=g\).

Short Answer

Expert verified
The set \(\{x \in X \mid f(x)=g(x)\}\) is closed; if equal on a dense subset, \(f=g\) on all \(X\).

Step by step solution

01

Understand the Concept of Closed Set in Topological Spaces

A set is closed in a topological space if its complement is open. In a Hausdorff space, we have additional properties regarding separation of points with disjoint open neighborhoods.
02

Define the Set of Interest

We need to show that the set \( \{ x \in X \mid f(x) = g(x) \} \) is closed in \(X\). This means we need to demonstrate that for any point \(x\) not in this set, it has an open neighborhood completely contained outside the set.
03

Leverage the Continuity of Functions

Since \(f\) and \(g\) are continuous functions from \(X\) to the Hausdorff space \(Y\), the inverse image of any closed set in \(Y\) is closed in \(X\). The set \( \{y \mid y = y\} \) in \(Y\) (i.e., the diagonal set \( \Delta = \{(y, y) \mid y \in Y\}\)) is closed because \(Y\) is Hausdorff. Therefore, \(\{x \in X \mid f(x) = g(x)\} = (f, g)^{-1}(\Delta)\) is closed.
04

Understand Dense Subset

A subset \(D\) of \(X\) is dense in \(X\) if its closure is \(X\). This means every point in \(X\) is either in \(D\) or a limit point of \(D\).
05

Show \(f = g\) on Dense Subset Extends to All of X

If \(f\) and \(g\) agree on a dense subset \(D\), let \(x \in X\). For each neighborhood \(U\) of \(x\), there exists a point \(d \in D \cap U\). Since \(f = g\) on \(D\), \(f(d) = g(d)\). Because \(f\) and \(g\) are continuous, and \(d\) can be chosen to be arbitrarily close to \(x\), we can conclude \(f(x) = g(x)\) for all \(x\in X\).
06

Conclusion

With the above steps, we've shown that the set \(\{x \in X \mid f(x) = g(x)\}\) is closed in \(X\), and if \(f = g\) on a dense subset of \(X\), then \(f = g\) on the entirety of \(X\).

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.

Topological Spaces
Topological spaces are a fundamental concept in mathematics, particularly in the area of topology. They provide a way to talk about the idea of space and continuity more generally. A topological space is defined by a set of points, along with a set of 'open' sets that determine the topological structure of that space. The open sets must satisfy three conditions:
  • The whole set and the empty set are included as open sets.
  • The union of any collection of open sets is also an open set.
  • The intersection of any finite number of open sets is also an open set.
These concepts help us understand the nearness of points without relying on a specific measurement of distance. Topological spaces may seem abstract, but they are crucial in understanding concepts of convergence, continuity, and compactness that are detailed in many branches of mathematics.
Hausdorff Space
A Hausdorff space is a type of topological space with an important property: it allows for the separation of distinct points. In such spaces, given any two distinct points, there exist disjoint open sets that each contain one of the points. This property is very useful, for example, in ensuring the uniqueness of limits. Some key characteristics of Hausdorff spaces include:
  • Every sequence in a Hausdorff space has at most one limit.
  • Closed subsets in a Hausdorff space are precisely those that can be defined as complements of open sets.
  • Continuous image of a compact subset in a Hausdorff space remains compact.
These characteristics make Hausdorff spaces a useful and often preferred setting for many areas of analysis and geometry, where the ability to separate points is essential.
Closed Sets
In the realm of topological spaces, a closed set is essentially the opposite of an open set. The simplest way to define a closed set is by stating that its complement within the space is an open set. Understanding closed sets is significant because they highlight important properties of convergence and boundaries within spaces. One of the most vital attributes of closed sets is that they contain all their limit points. This means that if a sequence of points in the set converges to a limit within the space, then the limit will also be contained within the set. This property is used heavily in the proof of the given problem, where we use continuity and the Hausdorff property to show that the set where the functions are equal is indeed closed.
Dense Subset
A dense subset of a topological space is a subset whose closure is the entire space. This concept means that every point in the topological space is either in the dense subset itself or is a limit point of the subset. Dense subsets are crucial because they can be used to approximate or capture the entirety of the space using a smaller, perhaps simpler set. Consider the rational numbers as a dense subset of the real numbers. Between any two real numbers, you can find a rational number. Some properties of dense subsets include:
  • If a function agrees on a dense subset, often it will extend to the whole space under certain conditions.
  • In metric spaces, a set is dense if and only if every point in the space is a limit of a sequence from the dense set.
By utilizing the properties of dense subsets, we can deduce that if two continuous functions agree on a dense subset, they will agree on the entire topological space.

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

Let \(X, Y\), and \(Z\) be topological spaces and give \(X \times Y\) the product topology. Show that if a function \(f: X \times Y \rightarrow Z\) is continuous, then it is separately continuous in each variable [i.e., for each \(x\) in \(X\) the function \(y \rightarrow f(x, y)\) is continuous from \(Y\), to \(Z\), and similarly for each \(y\) in \(Y\) ]. Show by an example that the converse does not hold. Hint: Try \(f(x, y)=x y\left(x^{2}+y^{2}\right)^{-1 / 2}\) if \((x, y) \neq(0,0)\) and \(f(0,0)=0\).

An order isomorphism between two ordered sets \((X, \leq)\) and \((Y, \leq)\) is a bijective map \(\varphi: X \rightarrow Y\) such that \(x_{1} \leq x_{2}\) iff \(\varphi\left(x_{1}\right) \leq \varphi\left(x_{2}\right) .\) A segment of a well-ordered set \((X, \leq)\) is a subset of \(X\) of the form \(\min \\{x\\}\) for some \(x\) in \(X\), or \(X\) itself (the improper segment). Show that if \(X\) and \(Y\) are well-ordered sets, then either \(X\) is order isomorphic to a segment of \(Y\) (with the relative order) or \(Y\) is order isomorphic to a segment of \(X\). Hint: The system of order isomorphisms \(\varphi: X_{\varphi} \rightarrow Y_{\varphi}\) where \(X_{\varphi}\) and \(Y_{\varphi}\) are segments of \(X\) and \(Y\), respectively, is inductively ordered if we define \(\varphi \leq \psi\) to mean \(X_{\varphi} \subset X_{\psi}\) (which implies that \(\varphi=\psi \mid X_{\varphi}\) and thus \(Y_{\varphi} \subset Y_{\psi}\) ). Prove that for a maximal element \(\varphi: X_{\varphi} \rightarrow Y_{\varphi}\) either \(X_{\varphi}\) or \(Y_{\varphi}\) must be an improper segment.

(The Sorgenfrey line.) Give the set \(\mathbb{R}\) the topology \(\tau\) for which a basis consists of the half-open intervals \([y, z[\), where \(y\) and \(z\) range over R. Show that every basis set is closed in \(\tau\). Show that \((\mathbb{R}, \tau)\) is a separable space that satisfies the first but not the second axiom of countability. Hint: If \(\rho\) is some basis for \(\tau\) and \(x \in \mathbb{R}\), then \(\rho\) must contain a set \(A\) such that \(x=\operatorname{Inf}\\{y \in A\\} .\)

Let \(X\) be an infinite set and let \(\tau\) denote the system of subsets \(A\) of \(X\) such that \(X \backslash A\) is finite, together with the set \(\emptyset .\) Show that \(\tau\) is a topology on \(X\), and that \((X, \tau)\) is compact, but not a Hausdorff space. Show that points are closed sets in \(X\). Show that every infinite subset of \(X\) is dense and deduce that \((X, \tau)\) is separable. Assume that \(X\) is uncountable and show that \((X, \tau)\) satisfies neither the first nor the second axiom of countability.

Let \(\mathbb{T}\) (torus) denote the unit circle in \(\mathbb{C}\), with the relative topology. Show that the product space \(\mathbb{T}^{2}\) (the 2-torus) is homeomorphic to a closed subset of \(\mathbb{R}^{3}\). Consider now the map \(f_{\theta}: \mathbb{R} \rightarrow \mathbb{T}^{2}\) given by \(f_{\theta}(x)=(\operatorname{expix}, \exp \mathrm{i} \theta x)\) for some \(\theta\) in \(\mathbb{R}\). Prove that \(f_{\theta}\) is continuous, and find a condition on \(\theta\) that makes \(f_{\theta}\) injective.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.