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Chapter 2: Problem 5

Let \(a\) be a point in a metric space \(X\). Let \(N\) be the set of positive integers. Prove that there is a collection \(\left\\{B_{n}\right\\}_{n \in N}\) of neighborhoods of \(a\) which constitutes a basis for the system of neighborhoods at \(a\).

Short Answer

Expert verified
The collection \(\{B_n\}_{n \in N}\) where \(B_n = \{x \in X \mid d(a, x) < \frac{1}{n}\}\) forms a basis for the neighborhoods at \(a\).

Step by step solution

01

Understanding the Problem

We are given a metric space \(X\) and a point \(a\) in \(X\). Our goal is to construct a collection of neighborhoods \(\{B_n\}_{n \in N}\) around the point \(a\), where \(N\) is the set of positive integers, that can serve as a basis for the neighborhood system of \(a\).
02

Define a Neighborhood in Metric Space

In a metric space, a neighborhood of a point \(a\) can be defined as an open ball \(B_{\epsilon}(a)\), where \(B_{\epsilon}(a) = \{x \in X \mid d(a, x) < \epsilon\}\) and \(d\) is the distance metric of the space. Here, \(\epsilon\) is a positive real number that controls the "size" of the ball.
03

Construct a Collection of Neighborhoods

We define a collection \(\{B_n\}_{n \in N}\) where each \(B_n\) is an open ball centered at \(a\) with radius \(\epsilon = \frac{1}{n}\). Therefore, \(B_n = \{x \in X \mid d(a, x) < \frac{1}{n}\}\) for each positive integer \(n\).
04

Verify the Basis Property

To prove that \(\{B_n\}\) forms a basis for the neighborhood system at \(a\), we need to show that for any neighborhood \(U\) of \(a\), there exists an \(n\) such that \(B_n \subseteq U\). Since \(U\) is a neighborhood of \(a\), there exists some \(\epsilon > 0\) such that \(B_\epsilon(a) \subseteq U\). Choose \(n\) such that \(\frac{1}{n} < \epsilon\), then \(B_n = B_{\frac{1}{n}}(a) \subseteq B_\epsilon(a)\) and thus \(B_n \subseteq U\).
05

Conclusion

Since we have shown that for any neighborhood \(U\) of \(a\), there is a neighborhood \(B_n\) from our collection that is contained in \(U\), \(\{B_n\}_{n \in N}\) forms a basis for the neighborhood system at \(a\). Hence, we have constructed the desired basis.

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Key Concepts

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

Open Ball
In the context of a metric space, an open ball is a fundamental concept used to define neighborhoods. An open ball centered at a point \(a\) with radius \(\epsilon\) in a metric space \(X\) is defined as the set of all points \(x\) within a specified distance from \(a\). Mathematically, this is represented as \(B_\epsilon(a) = \{x \in X \mid d(a, x) < \epsilon\}\), where \(d\) denotes the distance metric.

Key Points about Open Balls:
  • The center: The point \(a\) is the center of the open ball.
  • The radius: The value \(\epsilon\) is strictly a positive number and determines the size of the ball.
  • Contains points: Only points \(x\) such that their distance to \(a\) is less than \(\epsilon\) are included.
Open balls help us understand how 'close' or 'far' other points in the space are from the center \(a\). They are instrumental in defining other neighborhood and topology-related concepts.
Neighborhood System
A neighborhood system at a point \(a\) in a metric space is a collection of all neighborhoods of \(a\). This concept allows us to examine different surroundings of \(a\) within varying distances.

Characteristics of a Neighborhood System:
  • Variety of neighborhoods: Includes all open balls around \(a\) for different radii.
  • Control over distance: Each neighborhood provides a different local proximity around \(a\).
  • Uniqueness to each point: Different points in the space have unique neighborhood systems.
Knowing the neighborhood system of a point helps us in further analysis like continuity and limits by providing a broader understanding of closeness and proximity in a metric space.
Basis of Neighborhood System
A basis for the neighborhood system at a point \(a\) is a select collection of neighborhoods that can be used to generate all other neighborhoods at that point. This concept is akin to having a smaller, essential collection from which all other elements can be derived.

Important Aspects of a Basis:
  • Efficiency: Helps in constructing any neighborhood of the point without needing every possible open ball.
  • Inclusion property: For any neighborhood \(U\) of \(a\), there is at least one basis neighborhood that fits inside \(U\).
  • Simplicity: The basis is usually made up of open balls with systematically decreasing radii, like \(\{B_n\}_{n \in N}\) with radii \(\frac{1}{n}\) in this exercise.
The basis greatly simplifies the analysis of neighborhoods by reducing the complexity of dealing with an infinite number of possibilities.
Distance Metric
The distance metric in a metric space provides a way to measure the distance between any two points within that space. It's fundamental in defining the core structure of a metric space.

Essential Characteristics of a Distance Metric:
  • Non-negativity: For any two points \(a\) and \(b\), \(d(a, b) \geq 0\).
  • Identity: \(d(a, b) = 0\) if and only if \(a = b\).
  • Symmetry: The distance from \(a\) to \(b\) is the same as from \(b\) to \(a\), so \(d(a, b) = d(b, a)\).
  • Triangle inequality: For any points \(a, b, c\), \(d(a, b) + d(b, c) \geq d(a, c)\).
The distance metric is crucial as it validates the properties of open balls and their role in constructing neighborhood systems, ultimately guiding the behavior and relationships of points within a metric space.

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Most popular questions from this chapter

Let \(V\) be a vector space with the real numbers \(R\) as scalars. A function \(A\) : \(V\) \(\times V \rightarrow R\) is called a bilinear form if \(A(\alpha a+\beta b, c)=\alpha A(a, c)+\beta A(b, c)\) and \(A(a, \beta b+\gamma c)=\beta A(a, b)+\gamma A(a, c)\) for scalars \(\alpha, \beta\), and \(\gamma \in R\) and vectors \(a, b\), and \(c \in V\). A bilinear form is called positive definite if \(A(x\), \(x)>0\), unless \(x\) is the zero vector. Define a vector space structure on Hilbert space \(H\) and show that for \(u=\left(u_{1}, u_{2}, \ldots\right)\) and \(v=\left(v_{1}, v_{2}, \ldots\right) \in H, A(u, v)=\sum_{i=1}^{\infty} u_{i} v_{i}\) yields a positive definite bilinear form.

Let \(\left(Y, d^{\prime}\right)\) be a subspace of the metric space \((X, d)\). Prove that a subset \(O^{\prime} \subset Y\) is an open subset of \(\left(Y, d^{\prime}\right)\) if and only if there is an open subset \(O\) of \((X, d)\) such that \(O^{\prime}=Y \cap O\). Prove that a subset \(F^{\prime} \subset Y\) is a closed subset of \(\left(Y, d^{\prime}\right)\) if and only if there is a closed subset \(F\) of \((X, d)\) such that \(F^{\prime}=Y\) \(\cap F\). For a point \(a \in Y\), prove that a subset \(N^{\prime} \subset Y\) is a neighborhood of \(a\) if and only if there is a neighborhood \(N\) of \(a\) in \((X, d)\) such that \(N^{\prime}=Y \cap N\).

Let \(X\) be the set of all continuous functions \(f:[a, b] \rightarrow R\). For \(f, f, g \in X_{,}\) define $$ d(f, g)=\int_{a}^{b}|f(t)-g(t)| d t . $$ Using appropriate theorems from Calculus, prove that \((X, d)\) is a metric space.

Let \(a\) and \(b\) be distinct points of a metric space \(X\). Prove that there are neighborhoods \(N_{a}\) and \(N_{b}\) of \(a\) and \(b\) respectively such that \(N_{a} \cap N_{b}=\varnothing\).

Prove that the open interval \((-\pi / 2, \pi / 2)\), considered as a subspace of the real number system, is topologically equivalent to the real number system. Prove that any two open intervals, considered as subspaces of the real number system, are topologically equivalent. Prove that any open interval, considered as a subspace of the real number system, is topologically equivalent to the real number system.
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