Question

If \(f\) is a continuous mapping of a metric space \(X\) into a metric space \(Y\), prove that $$ f(E) \subset \overline{f(E)} $$ for every set \(E \subset X .\) (E denotes the closure of \(E\).) Show, by an example, that \(f(E)\) can be a proper subset of \(\overline{f(E)}\).

Short answer

Provide an example to demonstrate your answer. Answer: Yes, for any subset \(E \subset X\), the image of the subset under \(f\), denoted as \(f(E)\), always lies in the closure of its image, denoted as \(\overline{f(E)}\). An example is the sine function, \(f(x) = \sin(x)\), with \(X = Y = \mathbb{R}\) and \(E=[0, 2\pi]\). In this case, \(f(E)\) is a proper subset of \(\overline{f(E)}\) since \(f(E) = [-1, 1]\) and its closure, \(\overline{f(E)} = [-1, 1]\).

Step-by-step solution

Preview definitions and properties

Recall the definition of a continuous mapping: A function \(f\) is continuous at a point \(x\) if for every open neighborhood \(V\) of \(f(x)\), the inverse image \(f^{-1}(V)\) is an open neighborhood of \(x\). The closure of a set \(A\) in a metric space is the smallest closed set containing \(A\), which is equivalent to the union of the set \(A\) and its limit points, denoted as \(\overline{A}\).

Prove \(f(E) \subset \overline{f(E)}\)

We'll start by showing that for any subset \(E \subset X\), \(f(E) \subset \overline{f(E)}\). Let \(y \in f(E)\). Then there exists an \(x \in E\) such that \(f(x)=y\). Since \(f\) is continuous at the point \(x\), and \(f(E) \subset Y\), for every open neighborhood \(V\) of \(f(x)\) in \(Y\), the inverse image \(f^{-1}(V)\) is an open neighborhood of \(x\) in \(X\). As \(x \in E\), this means that \(x \in E \cap f^{-1}(V)\). Given that \(f\) is continuous and \(f(x)=y\), there must be a sequence \(\{x_n\}\) in \(E \cap f^{-1}(V)\) that converges to \(x\). As a result, the sequence \(\{f(x_n)\}\) converges to \(f(x)\). Therefore, \(y=f(x) \in \overline{f(E)}\) by definition, since \(f(x)\) is a limit point of \(f(E)\). Thus, we have shown that for any element \(y\) in \(f(E)\), \(y \in \overline{f(E)}\). Therefore, given any subset \(E\) of \(X\), we have obtained the desired result, \(f(E) \subset \overline{f(E)}\).

Show an example where \(f(E)\) is a proper subset of \(\overline{f(E)}\)

Consider the metric spaces \(X = Y = \mathbb{R}\) with the usual Euclidean metric, and let \(f(x) = \sin(x)\). The sine function is continuous on \(\mathbb{R}\). Now, let \(E = [0, 2\pi]\). Then \(f(E) = [-1, 1]\), since the sine function achieves its full range in \([0, 2\pi]\). The closure of \(f(E)\) is \(\overline{f(E)} = [-1, 1]\). In this case, \(f(E)\) is a proper subset of \(\overline{f(E)}\), since the interior of \(f(E)\) is \((-1, 1)\), and not equal to its closure \(\overline{f(E)} = [-1, 1]\). This example illustrates that \(f(E)\) can be a proper subset of \(\overline{f(E)}\).

Key concepts

Continuous Mapping

Understanding continuous mapping is fundamental in metric spaces. A continuous mapping, or function, from a metric space \(X\) into a metric space \(Y\) is one where small changes in \(X\) result in small changes in \(Y\). In formal terms, a function \(f\) is continuous at a point \(x \in X\) if, for every open neighborhood \(V\) of \(f(x)\), there exists an open neighborhood \(U\) of \(x\) such that \(f(U) \subseteq V\).
When considering the continuous mapping across the entire set, we say \(f\) is continuous over \(X\) if it is continuous at every point \(x \in X\). This concept helps ensure that the image of connected sets is connected, and it lays the groundwork for many important properties and theorems in analysis. Being familiar with the properties of continuous functions helps us understand how values in different metric spaces relate to each other.

Closure of a Set

The closure of a set \(A\), denoted as \(\overline{A}\), is a pivotal concept in topology and metric spaces. It is defined as the smallest closed set containing \(A\). To visualize closure, think of it as filling in every gap that might exist around or within the set \(A\).
Mathematically, the closure \(\overline{A}\) comprises all the points in \(A\) and all limit points of \(A\). A limit point of \(A\) is a point that can be approached by other points within \(A\) but isn't necessarily in \(A\) itself.

• If \(A\) is already closed, then \(\overline{A} = A\).
• Closure is crucial for understanding convergence and limit processes.

​In practical terms, if you're considering the closure of a set in a real-world scenario, it implies ensuring that no points are missed that are potentially part of the intended scope.

Euclidean Metric

The Euclidean metric, also known as the Euclidean distance, is a measure of distance in Euclidean space. Essentially, it is the straight-line distance between two points. If you have two points \((x_1, y_1)\) and \((x_2, y_2)\) in a 2D plane, the Euclidean distance between them is given by the formula: \[ d((x_1, y_1), (x_2, y_2)) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This concept generalizes to more dimensions, where the Euclidean distance formula extends to three dimensions using: \[ d((x_1, y_1, z_1), (x_2, y_2, z_2)) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]The Euclidean metric is intuitive and forms the basis for many mathematical and real-world applications.

• It's consistent with our everyday understanding of distance.
• It helps in defining open and closed balls in metric spaces.

Understanding the Euclidean metric is foundational to grasping more complex metric space concepts.

Limit Points

Limit points, or accumulation points, play a crucial role in topology and analysis. A limit point of a set \(A\) within a metric space is a point \(x\) where every open neighborhood around \(x\) contains at least one point from \(A\) different from \(x\) itself. This implies that the points in \(A\) "get arbitrarily close" to \(x\).
The idea of limit points is significant when discussing convergence, closed sets, and continuity:

• A set is closed if and only if it contains all its limit points.
• In sequences, a limit point corresponds to the value a sequence converges to.

Recognizing limit points enables one to understand the "boundaries" or "edges" of a set, even if those points are not explicitly part of the set. This helps in discerning characteristics of sequences, series, and functional limits. Understanding these concepts provides clarity in rigorous mathematical analysis.