Question

A set is said to be totally disconnected iff its only connected subsets are one-point sets and $\mathrm{\varnothing }$. Show that $R$ (the rationals) has this property in $E^1$.

Short answer

Rationals are totally disconnected in $E^1$ because every pair of rational numbers can be separated by irrationals.

Step-by-step solution

Define Intervals in Rationals

Firstly, understand that a connected subset of a set can not be broken into two disjoint nonempty open subsets. For the rationals $\mathbb{Q}$, consider a generic interval $(a,b)$.

Recognize Gaps in Rationals

Notice that between any two rational numbers, there exists an irrational number. This implies a gap in $\mathbb{Q}$, as the rationals within $(a,b)$ allow for splitting into disjoint open sets separated by irrationals.

Illustrate Separation with Irrationals

If you take any two rational numbers $x,y$ where $x<y$, you can find an irrational number $\alpha$ such that $x<\alpha <y$. Therefore, $\mathbb{Q}$ in $(a,b)$ can be expressed as a union of $(a,\alpha )\cap \mathbb{Q}$ and $(\alpha ,b)\cap \mathbb{Q}$, which are separate subsets.

Conclude Total Disconnection of $\mathbb{Q}$

Since any pair of rational numbers can be separated by an irrational in the interval $(a,b)$, no subset of $\mathbb{Q}$ other than singletons can remain connected. Thus, the rationals $\mathbb{Q}$ only have one-point connected subsets, fulfilling the condition of total disconnection.

Key concepts

Connected Subsets

In mathematics, understanding what it means for a subset to be "connected" is pivotal in studying topology or any field involving continuity and structure. A subset is considered connected if it is impossible to partition it into two disjoint nonempty open subsets. This means that the subset is in one piece without any breaks.

For any set, if you can never split it into two parts that are open and apart from each other, it maintains a sense of unity or connectedness. However, if you can find a method to divide it into such open and separate parts, then it is not connected. In our exploration of the rational numbers, we notice that, due to the presence of irrational numbers, $$\text{a connected subset of }\mathbb{Q}\text{ can only be a single point or an empty set.}$$

This distinct property is what makes the rational numbers a prime example of totally disconnected sets.

Rational Numbers

Rational numbers, denoted as $\mathbb{Q}$, are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Examples include $\frac{1}{2}$, $-3$, and $\frac{7}{4}$. These numbers are evenly spread out along the number line but do not include every number, leading to "gaps."

When examining intervals on the rational number line, these gaps become evident. A rational number interval, like $(a,b)$, includes all rationals between $a$ and $b$, but not necessarily every real number. This unique distribution affects their connectivity.

Because between any two rational numbers, an irrational number can be found, you can never form a connected whole using a range of rationals without running into a non-rational number. This persistent presence of irrationals ensures that rational subsets cannot join without being interrupted.

Irrational Numbers

Irrational numbers are numbers that cannot be written as a simple fraction. Their decimal expansions are infinite and non-repeating. Examples include $\pi$, $\sqrt{2}$, and $e$.

One pivotal role of irrational numbers in this study is their positioning within the real number line. They fill the gaps between rational numbers within any interval.

For instance, if you choose two rationals, like $x$ and $y$, there's always at least one irrational number, say $\alpha$, that lies between them, i.e., $x<\alpha <y$. This guarantees that, contrary to forming a continuous stretch of rationals, an interruption by irrationals is inevitable, causing any interval solely within rationals to be disjoint when considered as a part of all real numbers.

Intervals

An interval is a range on the number line that includes all numbers between two endpoints. Intervals can be open (excluding endpoints), closed (including endpoints), or half-open. Examples are $(a,b)$, $[a,b]$, and $(a,b]$.

In the context of rational numbers, any attempt to create a continuous stretch within an interval is thwarted by the inclusion of irrationals.

Even if you start with a rational interval $(a,b)$, the continuous stretch is broken. Because irrational numbers reside densely within these intervals, they effectively split $(a,b)$ into separate parts that do not connect inside $\mathbb{Q}$.

• No matter how small or large, any interval in rationals is subject to being split by irrationals.
• This splitting results in totally disconnected behavior as no continuous subset can be formed solely with rationals.

This insight helps appreciate the nuanced and disconnected nature of the rational numbers within real number space.