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Chapter 8: Problem 9

Prove: A limit point of a set \(S\) is either an interior point or a boundary point of \(S\).

Short Answer

Expert verified
Question: In a metric space, show that any limit point of a set S is either an interior point or a boundary point of S. Answer: If a point x is a limit point of a set S but not an interior point, it means that there does not exist an open ball centered at x entirely contained within S. In this case, every open ball centered at x must contain infinitely many points from S and points not in S, which is the definition of a boundary point. Thus, any limit point of S is either an interior point or a boundary point.

Step by step solution

01

Define limit point, interior point, and boundary point

First, let's define these terms: - Limit point: A point \(x\) is a limit point of a set \(S\) if every open ball centered at \(x\) contains infinitely many points from \(S\). - Interior point: A point \(x\) is an interior point of a set \(S\) if there exists an open ball centered at \(x\) that is entirely contained within \(S\). - Boundary point: A point \(x\) is a boundary point of a set \(S\) if every open ball centered at \(x\) contains both points of \(S\) and points not in \(S\). Now let's try to show that any limit point of a set \(S\) must either be an interior point or a boundary point of \(S\).
02

Suppose the limit point is not an interior point

Let's suppose that \(x\) is a limit point of the set \(S\) but is not an interior point. This means that there does not exist an open ball centered at \(x\) that is entirely contained within \(S\).
03

Show that the limit point must be a boundary point

Since \(x\) is a limit point and not an interior point, it follows that every open ball centered at \(x\) must contain infinitely many points from \(S\). However, as stated earlier, there does not exist an open ball centered at \(x\) that is entirely contained within \(S\). Thus, every open ball centered at \(x\) must also contain points not in \(S\). This is precisely the definition of a boundary point. So, we have shown that if \(x\) is a limit point of \(S\) and not an interior point, then it must be a boundary point. Therefore, a limit point of a set \(S\) is either an interior point or a boundary point of \(S\).

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Most popular questions from this chapter

Show that if \(\mathbf{X}=\left\\{x_{i}\right\\}_{i=1}^{\infty} \in \ell_{p}\) and \(\mathbf{Y}=\left\\{y_{i}\right\\}_{i=1}^{\infty} \in \ell_{q},\) where \(1 / p+1 / q=1,\) then \(\mathbf{Z}=\left\\{x_{i} y_{i}\right\\} \in \ell_{1}\)

Let \(\left\\{T_{n}\right\\}_{n=1}^{\infty}\) be a sequence of nonempty closed sets of a metric space such that (a) \(T_{1}\) is compact; (b) \(T_{n+1} \subset T_{n}, n \geq 1 ;\) and (c) \(\lim _{n \rightarrow \infty} d\left(T_{n}\right)=0 .\) Show that \(\cap_{n=1}^{\infty} T_{n}\) contains exactly one member.

Suppose that \(\mathbf{X}=\left\\{x_{i}\right\\}_{i=1}^{\infty}\) is in \(\ell_{p},\) where \(p>1\). Show that (a) \(\mathbf{X} \in \ell_{r}\) for all \(r>p\) (b) If \(r>p,\) then \(\|\mathbf{X}\|_{r} \leq\|\mathbf{X}\|_{p}\) (c) \(\lim _{r \rightarrow \infty}\|\mathbf{X}\|_{r}=\|\mathbf{X}\|_{\infty}\).

(a) Show that $$ \|f\|=\int_{a}^{b}|f(x)| d x $$ is a norm on \(C[a, b]\), (b) Show that the sequence \(\left\\{f_{n}\right\\}\) defined by $$ f_{n}(x)=\left(\frac{x-a}{b-a}\right)^{n} $$ is a Cauchy sequence in \((C[a, b],\|\cdot\|)\) (c) Show that ( \(C[a, b],\|\cdot\|)\) is not complete.

For each positive integer \(i,\) let \(\left(A_{i}, \rho_{i}\right)\) be a metric space. Let \(A\) be the set of all objects of the form \(\mathbf{X}=\left(x_{1}, x_{2}, \ldots, x_{n}, \ldots\right),\) where \(x_{i} \in A_{i}, i \geq 1 .\) (For example, if \(A_{i}=\mathbb{R}, i \geq 1,\) then \(A=\mathbb{R}^{\infty}\).) Let \(\left\\{\alpha_{i}\right\\}_{i=1}^{\infty}\) be any sequence of positive numbers such that \(\sum_{i=1}^{\infty} \alpha_{i}<\infty\) (a) Show that $$ \rho(\mathbf{X}, \mathbf{Y})=\sum_{i=1}^{\infty} \alpha_{i} \frac{\rho_{i}\left(x_{i}, y_{i}\right)}{1+\rho_{i}\left(x_{i}, y_{i}\right)} $$ is a metric on \(A\). (b) Let \(\left\\{\mathbf{X}_{r}\right\\}_{r=1}^{\infty}=\left\\{\left(x_{1 r}, x_{2 r}, \ldots, x_{n r}, \ldots\right)\right\\}_{r=1}^{\infty}\) be a sequence in \(A .\) Show that $$ \lim _{r \rightarrow \infty} \mathbf{X}_{r}=\widehat{\mathbf{X}}=\left(\widehat{x}_{1}, \widehat{x}_{2}, \ldots, \widehat{x}_{n}, \ldots\right) $$ if and only if $$ \lim _{r \rightarrow \infty} x_{i r}=\widehat{x}_{i}, \quad i \geq 1 $$ (c) Show that \(\left\\{\mathbf{X}_{r}\right\\}_{r=1}^{\infty}\) is a Cauchy sequence in \((A, \rho)\) if and only if \(\left\\{x_{i r}\right\\}_{r=1}^{\infty}\) is a Cauchy sequence in \(\left(A_{i}, \rho_{i}\right)\) for all \(i \geq 1\). (d) Show that \((A, \rho)\) is complete if and only if \(\left(A_{i}, \rho_{i}\right)\) is complete for all \(i \geq 1\).
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