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Chapter 4: Problem 1

On the real line, prove that the set of non-zero numbers is not a connected set.

Short Answer

Expert verified
The set \\(\mathbb{R} \setminus \{0\}\\) is not connected because it can be split into two disjoint open sets \\( (-\infty,0) \\) and \\( (0,\infty) \\).

Step by step solution

01

Understanding Connectivity

A set in the real line is connected if it cannot be divided into two disjoint open sets that together form the original set.
02

Defining the Set

The set of non-zero numbers on the real line is defined as \({\mathbb{R} \setminus \{0\}}\). This set includes all real numbers except zero.
03

Consider Open Sets Surrounding Zero

Consider the two open sets: \(A = (-\infty, 0)\) and \(B = (0, \infty)\). Together, these sets cover all non-zero real numbers, i.e., \({\mathbb{R} \setminus \{0\} = A \cup B}\), and are disjoint.
04

Verify Disjointness

The intersection of sets \(A\) and \(B\) is empty, \(A \cap B = \emptyset\), which confirms that they are disjoint.
05

Conclude Non-Connectivity

Since we can express \({\mathbb{R} \setminus \{0\}}\) as the union of two disjoint open sets, it is not connected. A connected set cannot be split into two disjoint open sets in this manner.

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Key Concepts

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

Connected Set
In topology, a set is referred to as "connected" if it cannot be divided into two separate open sets that together form the original set.
In simpler terms, this means that we cannot find a way to split the set into non-overlapping parts while still keeping everything inside the original set.
  • Being connected implies a certain level of "wholeness" or "continuity."
  • A connected set does not have gaps which can allow it to be broken into isolated parts.
This concept is crucial in understanding many areas of mathematics, as it illustrates how some sets are inherently linked together, while others can be divided.
Real Line
The real line, often denoted by \({\mathbb{R}}\), includes all real numbers and can be visualized as an infinite line containing every possible point that represents a real number.
  • The real line is continuous, without any breaks between the numbers.
  • It extends infinitely in both positive and negative directions.
The real line also serves as the foundation for other mathematical concepts, including intervals and limits, offering a complete numerical representation of reality.
Open Sets
In topology, an open set is a concept that defines a set of points, where every point is surrounded by a neighborhood entirely contained within the set.
  • Open sets are essential for defining continuous functions and spaces.
  • On the real line, open intervals like \((a, b)\) are classical examples of open sets.
Understanding open sets allows us to grasp how certain spaces can be explored without encountering boundaries, enabling fluid transitions from one point to another within the set.
Disjoint Sets
Disjoint sets are collections of elements that have no elements in common, meaning their intersection is the empty set, or \(\emptyset\).
  • If two sets are disjoint, any element that belongs to one set does not belong to the other.
  • For example, the sets \(A = ( -\infty, 0)\) and \(B = (0, +\infty)\) are disjoint as they do not share any common elements.
In topology, disjoint sets are significant because they help identify spaces or degrees of separation within larger entities, aiding in the analysis of connectedness and continuity.

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Most popular questions from this chapter

Verify that in a topological space \(X\) (i) if there is a path with initial point \(A\) and terminal point \(B\), then there is a path with initial point \(B\) and terminal point \(A\), and (ii) if there is a path connecting points \(A\) and \(B\) and a path connecting points \(B\) and \(C\), then there is a path connecting points \(A\) and \(C\).

Prove directly by constructing appropriate paths that the topological spaces \(R^{n}, I^{n}\) (the unit cube), and \(S^{n}(n>0)\) are pathconnected.

Let \(F: R^{2} \rightarrow R\) be a real-valued function defined and continuous on the plane. For each continuous function \(f:[a, b] \rightarrow R\) we may define a new continuous function \(K f:[a, b] \rightarrow R\) by setting \(K f(t)=\int_{a}^{t} F(x, f(x)) d x, t \in[a, b]\). Thus, if \(S\) is the set of continuous realvalued functions defined on \([a, b], K\) defines a transformation of \(S\) into itself. Prove that an element \(g \in S\) is a fixed point of \(K\) if and only if \(g\) satisfies the differential equation \(g^{\prime}(x)=F(x, g(x))\) with initial condition \(g(a)=0\).

Prove that for each positive integer \(n, R^{n}\) and \(I^{n}\) are simply connected.

Let \(\left\\{X_{\alpha}\right\\}_{a \in A}\) be an indexed family of topological spaces and set \(X=\Pi_{\alpha \in A} X_{\alpha}\). For each \(\alpha \in A\) let \(f_{\alpha}: I \rightarrow X_{\alpha}\) be a path in \(X_{\alpha}\). Set \(\left(f_{A}(t)\right)(\alpha)=\) \(f_{\alpha}(t)\) so that \(f_{A}: I \rightarrow X\). Prove that \(f_{A}\) is a path in \(X\). Prove that if each \(X_{\alpha}\) is path-connected, so is \(X\).
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