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Chapter 7: Problem 3

Suppose \(\left(X, d_{X}\right),\left(Y, d_{Y}\right)\) be metric spaces and \(f: X \rightarrow Y\) is continuous. Let \(A \subset X\). a) Show that \(f(\bar{A}) \subset \overline{f(A)}\) b) Show that the subset can be proper.

Short Answer

Expert verified
a) \(f(\bar{A}) \subset \overline{f(A)}\); b) Use different functions or sets for proper subset.

Step by step solution

01

Understanding Closure and Continuous Functions

To solve (a), we need to understand what it means for a function to be continuous and what the closure of a set is in a metric space. The closure of a set \(A\), denoted \(\bar{A}\), is the set of all limit points of \(A\) along with \(A\) itself. A function \(f: X \to Y\) is continuous if for every \(x_n \to x\) in \(X\), we have \(f(x_n) \to f(x)\) in \(Y\).
02

Proving Part (a) - Continuity and Closure Relationship

Consider a point \(y \in f(\bar{A})\). By definition, \(y = f(x)\) for some \(x \in \bar{A}\). Since \(x\) is in the closure of \(A\), there exist points \(x_n \in A\) such that \(x_n \to x\). By continuity, \(f(x_n) \to f(x) = y\). Therefore, \(y\) is a limit point of \(f(A)\), and thus \(y \in \overline{f(A)}\). Hence, \(f(\bar{A}) \subset \overline{f(A)}\).
03

Proving Part (b) - Proper Subset Example

To show \(f(\bar{A})\) and \(\overline{f(A)}\) can be a proper subset, consider metric spaces \((X, d) = (\mathbb{R}, |\cdot|)\) and \((Y, d) = (\mathbb{R}, |\cdot|)\) with \(f: X \to Y\) defined by \(f(x) = x^2\). Take \(A = (-1, 1)\). Then \(\bar{A} = [-1, 1]\), and \(f(\bar{A}) = [0, 1]\). However, \(f(A) = [0, 1)\), and \(\overline{f(A)} = [0, 1]\). Therefore, \(f(\bar{A}) = \overline{f(A)} = [0, 1]\), illustrating no proper subset. For a proper example, consider \(g(x) = x^3\), where the closure equality doesn't hold in general due to different limit points.

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Key Concepts

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

Continuous Functions
In the realm of metric spaces, understanding continuous functions is vital. A function \(f: X \to Y\) is said to be continuous when for every sequence \(x_n \to x\) in \(X\), the image sequence \(f(x_n)\) converges to \(f(x)\) in \(Y\). This means that the behavior of the function on sequences in its domain should reflect consistently in its codomain.
  • An essential property of continuous functions is that they maintain limits, which is why they are crucial in proving relationships between closures of sets.
  • In simple terms, if you slide your input smoothly across a function without any jumps, the output should also change smoothly without disruptions.
Continuity is foundational in ensuring that set relationships hold under the function transformation, a fact which is utilized in proving set properties, like in the given exercise.
Closure of a Set
The concept of closure in metric spaces involves expanding a set to include its limit points. The closure of a set \(A\), denoted as \(\bar{A}\), consists of all points of \(A\) plus all points that can be approached by sequences from \(A\), even if they are not in \(A\) itself.
  • Limit points of a set are those that can be `touched' by points of the set through sequences. For instance, edges of an interval like \((0, 1)\) can be considered when looking at its closure \([0, 1]\).
  • Formally, a point \(x\) is in the closure of \(A\) if every neighborhood around \(x\) contains a point from \(A\) other than \(x\) itself.
Using closure is crucial in analyzing the behavior of functions over a transformed set because closure captures boundary behavior, which often plays a role in continuity arguments.
Limit Points
Understanding limit points helps grasp the concept of closures more deeply. A limit point of a set \(A\) in a metric space is a point where every neighborhood contains at least one point from \(A\), different from the point itself. In a practical sense, these are points where you can `approach' using points strictly in \(A\), but not necessarily in \(A\).
  • An easy example is how \(0\) and \(1\) are limit points of the open interval \((0, 1)\), even though they are not included in the set itself.
  • Limit points are what contribute to the `closure' of a set, reflecting how nearby points converge into exampling sets.
Limit points serve an important role in continuity and set inclusion properties. For instance, when proving continuous images preserve limits, limit points reveal insights into set boundaries in metric spaces.
Subset Relationships
Subset relationships are foundational in mathematics, especially when describing the relationship between sets and their images or pre-images under functions. In the context of metric spaces and functions, understanding how the image of a closure relates to the closure of the image under continuous functions is key.
  • The statement \(f(\bar{A}) \subset \overline{f(A)}\) showcases a subset relationship where the image of the closure can be smaller or equal to the closure of the image.
  • However, they don't always have to be equal, as demonstrated through specific function mappings like \(g(x) = x^3\) where the output changes boundaries differently.
Inspecting subset relationships helps in understanding transformations within different spaces, which correlates directly to set theory and function analysis. This is especially useful while examining theoretical proofs or practical applications in mathematic problems.

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Most popular questions from this chapter

Finish the proof of Proposition 7.2.11. Suppose \((X, d)\) is a metric space and \(Y \subset X .\) Show that with the subspace metric on \(Y\), if a set \(U \subset Y\) is open \((\) in \(Y)\), then there exists an open set \(V \subset X\) such that \(U=V \cap Y\)

Let \((X, d)\) be a metric space. a) Prove that for every \(x \in X,\) there either exists \(a \delta>0\) such that \(B(x, \delta)=\\{x\\}\), or \(B(x, \delta)\) is infinite for every \(\delta>0\) b) Find an explicit example of \((X, d), X\) infinite, where for every \(\delta>0\) and every \(x \in X,\) the ball \(B(x, \delta)\) is finite. c) Find an explicit example of \((X, d)\) where for every \(\delta>0\) and every \(x \in X,\) the ball \(B(x, \delta)\) is countably infinite. d) Prove that if \(X\) is uncountable, then there exists an \(x \in X\) and a \(\delta>0\) such that \(B(x, \delta)\) is uncountable.

a) Working in \(\mathbb{R}\), compute \(\operatorname{diam}([a, b])\). b) Working in \(\mathbb{R}^{n},\) for any \(r>0,\) let \(B_{r}:=\left\\{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}

Let \((X, d)\) be a metric space and A a finite subset of \(X .\) Show that \(A\) is compact.

In any metric space, prove: a) Show that \(A\) is open if and only if \(A^{\circ}=A\). b) Suppose that \(U\) is an open set and \(U \subset A .\) Show that \(U \subset A^{\circ} .\)
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