R³ Connectivity: Exploring Sets With One Coordinate In A

Introduction

Hey guys! Today, we're diving deep into a fascinating question in general topology, a follow-up to a previous discussion about the connectedness of sets in $\mathbb{R}^3$. Specifically, we're going to investigate whether the set of points in three-dimensional space ($\mathbb{R}^3$) with exactly one coordinate belonging to a given set $A$ can be connected. This is a really cool concept that blends geometry and set theory, so buckle up and let's get started! We will explore this question thoroughly, providing detailed explanations and examples to ensure a comprehensive understanding of the topic. Our main goal is to determine the conditions under which such a set remains connected and the implications of these conditions in the realm of topology. Let's unravel the intricacies of this problem together!

Background: Revisiting the Rational Coordinate Case

Before we jump into the general case with set $A$, let's quickly recap the problem that sparked this question. The original problem asked whether the set of points in $\mathbb{R}^3$ with exactly one coordinate being a rational number ($\mathbb{Q}$) is connected. The answer, as it turns out, is no. This is a crucial starting point because it gives us a specific example to contrast with the general case. Understanding why the rational coordinate case fails to be connected will help us appreciate the subtleties involved in determining the connectedness of our target set. The disconnection arises from the nature of rational numbers, which are interspersed with irrational numbers, creating gaps that prevent continuous paths between certain points. This initial example serves as a valuable foundation for our subsequent analysis of the general case, where the set $A$ can have various properties that may or may not lead to connectedness.

The Core Question: Connectivity with One Coordinate in A

Now, let's formalize our main question. We're considering a set $S$ in $\mathbb{R}^3$ defined as follows: $S = \{(x, y, z) \in \mathbb{R}^3 \mid$ exactly one of $x$, $y$, or $z$ is in $A\}$, where $A$ is a subset of the real numbers ($\mathbb{R}$). The burning question is: Under what conditions on $A$ is the set $S$ connected? To tackle this, we need to think about what it means for a set to be connected. Intuitively, a set is connected if you can draw a continuous path between any two points in the set without leaving the set. Mathematically, a set is connected if it cannot be expressed as the union of two disjoint, non-empty open sets. Understanding connectedness is crucial for our investigation, as it forms the very basis of our inquiry. We will explore various scenarios and properties of the set $A$ to see how they impact the connectedness of $S$. So, let’s delve deeper into the conditions that might ensure the connectedness of $S$.

Exploring Conditions for Connectedness

To make progress, let's start by considering some specific examples of the set $A$. What if $A$ is an interval, say $[0, 1]$? Or what if $A$ is the set of integers $\mathbb{Z}$? These concrete examples can give us some intuition. If $A$ is an interval, it seems plausible that $S$ might be connected. If $A$ is the set of integers, we might anticipate that $S$ is not connected, similar to the rational number case. Exploring these examples is a powerful technique in mathematics. By examining specific cases, we can often identify patterns and develop hypotheses about the general case. Let’s break down how different properties of $A$ might influence the connectedness of $S$. For instance, the density of $A$ in $\mathbb{R}$ (meaning that between any two real numbers, there's an element of $A$) might play a significant role. Also, the presence of isolated points in $A$ could be a factor in disconnecting $S$.

The Role of Path-Connectedness

It's often easier to work with the concept of path-connectedness, which is a stronger condition than connectedness. A set is path-connected if any two points in the set can be joined by a continuous path that lies entirely within the set. If we can show that $S$ is path-connected, then it is also connected. So, let's try to construct a path between two arbitrary points in $S$. Suppose we have two points $P_1 = (x_1, y_1, z_1)$ and $P_2 = (x_2, y_2, z_2)$ in $S$. Exactly one coordinate of each point is in $A$. To connect these points, we need to find a continuous path within $S$. This often involves carefully manipulating the coordinates so that only one coordinate at a time moves through the set $A$ or its complement. Path-connectedness provides a tangible way to approach the problem. By focusing on constructing paths, we can develop a clearer understanding of the connectivity properties of $S$. The challenge lies in ensuring that the constructed path remains within the set $S$, meaning that at any point along the path, exactly one coordinate is in $A$.

Cases Where A is Path-Connected

If $A$ is path-connected (for example, an interval), we can often construct a path between any two points in $S$. Suppose $x_1 o x_2$ is the coordinate in A. We can keep y and z fixed and only allow x to vary. Then we repeat for the next coordinate. However, when $A$ is path-connected, constructing a path between points in $S$ might be more straightforward. For instance, if $A$ is an interval, we can use the connectedness of the interval to move the coordinates smoothly from one point to another while ensuring that only one coordinate at a time belongs to $A$. The path-connectedness of $A$ simplifies the problem significantly. When $A$ is path-connected, we have a continuous medium within $A$ that allows us to move coordinates smoothly. This often translates into the path-connectedness of $S$. However, we need to consider cases where $A$ might not be path-connected, and how that affects the connectivity of $S$.

Disconnectedness When A is Not Well-Behaved

If $A$ is a set like the rational numbers, the story changes. The discontinuities in $A$ can lead to $S$ being disconnected. Imagine trying to move a coordinate continuously through the rationals without hitting an irrational number. It's impossible! This intuition aligns with our earlier observation that the set of points in $\mathbb{R}^3$ with exactly one rational coordinate is disconnected. The “well-behavedness” of $A$ is crucial for the connectedness of $S$. When $A$ is not well-behaved, such as when it has many gaps or isolated points, the connectedness of $S$ is jeopardized. The discontinuities in $A$ prevent us from constructing continuous paths between points in $S$. This highlights the interplay between the properties of $A$ and the resulting topological properties of $S$.

Key Results and Considerations

Here's a key takeaway: If $A$ is a connected subset of $\mathbb{R}$, then $S$ is likely connected. Conversely, if $A$ is disconnected, then $S$ is likely disconnected. However, this is just a rule of thumb, and we need to be careful with the word