Problem 8
Question
(The Sorgenfrey plane.) Give the set \(\mathbb{R}^{2}\) the topology \(\tau^{2}\), for which a basis consist of products of half-open intervals \(\left[y_{1}, z_{1}\left[\times\left[y_{2}, z_{2}[\right.\right.\right.\), where \(y_{1}, y_{2}, z_{1}\), and \(z_{2}\) range over \(\mathbb{R}\). Show that \(\left(\mathbb{R}^{2}, \tau^{2}\right)\) is a separable space. Show that the subset \(\left\\{(x, y) \in \mathbb{R}^{2} \mid x+y=0\right\\}\) is discrete in the relative topology (and thus nonseparable), but closed in \(\mathbb{R}^{2}\).
Step-by-Step Solution
Verified Answer
$\mathbb{R}^2$ is separable; the set $S = \{(x, y) \mid x+y=0\}$ is discrete in the subspace topology and closed in $\mathbb{R}^2$.
1Step 1: Understanding the topology on $\mathbb{R}^2$
The Sorgenfrey plane topology \(\tau^2\) on \(\mathbb{R}^{2}\) consists of a basis formed by products of half-open intervals of the form \([y_1, z_1) \times [y_2, z_2)\), with \(y_1, y_2, z_1\), and \(z_2\) being real numbers.
2Step 2: Showing that $(\mathbb{R}^2, \tau^2)$ is separable
To show separability, we need a countable dense subset of \(\mathbb{R}^2\). Consider the subset \(D = \{ (q_1, q_2) \mid q_1, q_2 \in \mathbb{Q} \}\), where \(\mathbb{Q}\) is the set of rational numbers. Given any basic open set \(U = [y_1, z_1) \times [y_2, z_2)\), choose rational numbers \(q_1, q_2\) such that \(y_1 \leq q_1 < z_1\) and \(y_2 \leq q_2 < z_2\). Hence, \(D \cap U eq \emptyset\). Thus, \(D\) is dense in \(\mathbb{R}^2\).
3Step 3: Identifying and understanding the subset $S = \{ (x, y) \mid x + y = 0 \}$
The subset \(S\) is a line in \(\mathbb{R}^2\) consisting of all points where the sum of the coordinates is zero. In other words, \(S\) can be described as \(S = \{ (x, -x) \mid x \in \mathbb{R} \}\).
4Step 4: Showing $S$ is discrete in the subspace topology
To be discrete in the subspace topology, each point \((a, -a) \in S\) must have an open set in the relative topology that contains only \((a, -a)\). Consider the basic open set \(U = [a, a+\epsilon) \times [-a, -a+\epsilon)\) for some small \(\epsilon > 0\). Then \(U \cap S = \{ (a, -a) \}\), showing each point is isolated in \(S\), hence \(S\) is discrete.
5Step 5: Showing $S$ is closed in $(\mathbb{R}^2, \tau^2)$
To show \(S\) is closed, its complement must be open. The complement of \(S\) consists of points \((x, y)\) where \(x + y eq 0\). For any such point, take \(\epsilon = \frac{|x + y|}{2}\) and consider the open set \(V = [x - \epsilon, x) \times [y - \epsilon, y)\). For any point \((x', y') \in V\), \(x' + y' eq 0\), hence no points of \(S\) are in \(V\). Thus, the complement of \(S\) is open and \(S\) is closed.
Key Concepts
TopologySeparable spaceDiscrete subsetClosed set
Topology
In mathematics, topology is a branch that extends beyond the traditional concepts of geometry and set theory. It focuses on the properties of space that are preserved under continuous transformations such as stretching or bending,
but not tearing or gluing. Topology thrives on "open sets," which provides a framework to define concepts like continuity, compactness, and convergence.
In the context of the Sorgenfrey plane, we deal with a unique topology on \( \mathbb{R}^2 \), where the basis is defined by products of half-open intervals of the form \( [y_1, z_1) \times [y_2, z_2) \).
This special basis distinguishes the Sorgenfrey topology from the standard Euclidean topology and leads to interesting properties such as different notions of convergence and separability.
The Sorgenfrey plane provides an excellent example to study non-standard topologies and their implications in \( \mathbb{R}^2 \).
but not tearing or gluing. Topology thrives on "open sets," which provides a framework to define concepts like continuity, compactness, and convergence.
In the context of the Sorgenfrey plane, we deal with a unique topology on \( \mathbb{R}^2 \), where the basis is defined by products of half-open intervals of the form \( [y_1, z_1) \times [y_2, z_2) \).
This special basis distinguishes the Sorgenfrey topology from the standard Euclidean topology and leads to interesting properties such as different notions of convergence and separability.
The Sorgenfrey plane provides an excellent example to study non-standard topologies and their implications in \( \mathbb{R}^2 \).
Separable space
A topological space is called separable if it contains a countable, dense subset. This is a crucial concept in topology as it often simplifies the study of spaces by reducing infinite dimensional spaces to a countable framework.
In the Sorgenfrey plane \( (\mathbb{R}^2, \tau^2) \), we can identify a separable space
by considering the set \( D = \{ (q_1, q_2) \mid q_1, q_2 \in \mathbb{Q} \} \), where \( \mathbb{Q} \) represents rational numbers.
Rationals are dense in the real numbers, which means that any point can be "approached" arbitrarily closely by points from \( D \).
This trait makes \( D \) a countable dense subset. In the Sorgenfrey topology,
by constructing products of these rational numbers to form elements of the basis of open sets, we find that \( D \cap U eq \emptyset \), which demonstrates the density and hence the separability of the space.
This idea of separability often helps relate these spaces to more familiar structures, supporting analysis on such spaces.
In the Sorgenfrey plane \( (\mathbb{R}^2, \tau^2) \), we can identify a separable space
by considering the set \( D = \{ (q_1, q_2) \mid q_1, q_2 \in \mathbb{Q} \} \), where \( \mathbb{Q} \) represents rational numbers.
Rationals are dense in the real numbers, which means that any point can be "approached" arbitrarily closely by points from \( D \).
This trait makes \( D \) a countable dense subset. In the Sorgenfrey topology,
by constructing products of these rational numbers to form elements of the basis of open sets, we find that \( D \cap U eq \emptyset \), which demonstrates the density and hence the separability of the space.
This idea of separability often helps relate these spaces to more familiar structures, supporting analysis on such spaces.
Discrete subset
A subset of a topological space is termed discrete if every point in the subset is isolated,
meaning it can be "separated" from other points by surrounding each with its own exclusive open set.
In the Sorgenfrey plane, we examine the subset \( S = \{ (x, y) \mid x + y = 0 \} \), represented as the line \( S = \{ (x, -x) \mid x \in \mathbb{R} \} \).
To see it as a discrete subset in the Sorgenfrey topology, we look at the relative topology applied to \( S \).
For a point \( (a, -a) \in S \), we can define a basic open set
such as \( U = [a, a+\epsilon) \times [-a, -a+\epsilon) \), which only contains \( (a, -a) \). This isolating nature of each point establishes \( S \) as discrete.
This concept is essential in topology as discrete sets highlight how different the properties of certain spaces can be under different topologies.
meaning it can be "separated" from other points by surrounding each with its own exclusive open set.
In the Sorgenfrey plane, we examine the subset \( S = \{ (x, y) \mid x + y = 0 \} \), represented as the line \( S = \{ (x, -x) \mid x \in \mathbb{R} \} \).
To see it as a discrete subset in the Sorgenfrey topology, we look at the relative topology applied to \( S \).
For a point \( (a, -a) \in S \), we can define a basic open set
such as \( U = [a, a+\epsilon) \times [-a, -a+\epsilon) \), which only contains \( (a, -a) \). This isolating nature of each point establishes \( S \) as discrete.
This concept is essential in topology as discrete sets highlight how different the properties of certain spaces can be under different topologies.
Closed set
In topology, a set is considered closed if it contains all its limit points; equivalently,
its complement in the host space is open. For the Sorgenfrey plane, we show this with the subset \( S = \{ (x, y) \mid x + y = 0 \} \).
The complement of \( S \) in the Sorgenfrey plane consists of all points \( (x, y) \) where \( x + y e 0 \).
We prove openness by considering an open set \( V = [x - \epsilon, x) \times [y - \epsilon, y) \) for any point in the complement.
This set effectively excludes any point where \( x + y = 0 \). Resultantly,
because its complement consists of basic open sets in the Sorgenfrey topology, \( S \) is verified to be closed.
Understanding such properties helps in the broader study of space-related concepts such as boundaries and limits, which further ties into continuity and convergence discussions.
its complement in the host space is open. For the Sorgenfrey plane, we show this with the subset \( S = \{ (x, y) \mid x + y = 0 \} \).
The complement of \( S \) in the Sorgenfrey plane consists of all points \( (x, y) \) where \( x + y e 0 \).
We prove openness by considering an open set \( V = [x - \epsilon, x) \times [y - \epsilon, y) \) for any point in the complement.
This set effectively excludes any point where \( x + y = 0 \). Resultantly,
because its complement consists of basic open sets in the Sorgenfrey topology, \( S \) is verified to be closed.
Understanding such properties helps in the broader study of space-related concepts such as boundaries and limits, which further ties into continuity and convergence discussions.
Other exercises in this chapter
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