91Ó°ÊÓ

When working with real analysis, understanding open and closed intervals is essential. An **open interval** in the real numbers, denoted as \(I = (a, b)\), contains all numbers strictly between \(a\) and \(b\). This means that neither \(a\) nor \(b\) are included in the interval, hence the 'open'. For example, if you imagine all numbers between 1 and 5, without counting 1 and 5, you have an open interval.

On the other hand, a **closed interval**, represented as \(J = [c, d]\), encompasses all numbers between and including its endpoints \(c\) and \(d\). So, if you think of all numbers from 2 to 6, including 2 and 6 themselves, you have a closed interval.

• Open intervals: Points inside are not including the endpoints.
• Closed intervals: Points inside including the endpoints.

These definitions become crucial when mapping these intervals onto other spaces, such as \( \mathbb{R}^{2} \), where they determine the openness or closedness of related sets like \( I_1 \) and \( J_1 \).

To explore intervals in \( \mathbb{R}^{2} \), it's pivotal to grasp the idea of **subspace topology**. When we talk about topology, we address the properties of space that are preserved under continuous deformations such as stretching or bending. Now, imagine we restrict our interest just to a line, like \( y = 0 \), in the entire \( \mathbb{R}^{2} \). This is what is meant by a subspace.

In the subspace, the openness or closedness of a set is determined not by the flat plane of \( \mathbb{R}^{2} \), but by the line itself.

• A set that might be open in the full space \( \mathbb{R}^2 \) might not be open within a restricted line.
• Similarly, sets that appear closed in regular spaces may not hold the same criteria in subspaces.

For example, as with \( I_1 \) - when restricted to \( y=0 \), it lacks all necessary neighborhoods in \( \mathbb{R}^{2} \) thus is not open. However, \( J_1 \) remains closed because it includes all limit points and endpoints on the line.

**Limit points** are instrumental in assessing closedness. A limit point of a set is a point where, no matter how closely you zoom into it, you still find other points from the set clinging around. This means, any sequence in a set that converges to a point should also have this point within the set for it to be considered closed.

In the example, \( J_1 =[c, d] \) consists of all numbers from \( c \) to \( d \). Imagine a sequence approaching the right end, \( d \). Since \( J_1 \) contains its endpoints \(c,0\) and \(d,0\), and all other numbers between them, it is considered closed in the subspace topology.

• A set is closed if it contains all its limit points.
• Even the tiniest sequence from the set cannot escape it.

Hence, in the subspace topology on the x-axis \(J_1\) is closed, as its endpoints form part of the interval, unlike open interval \(I_1\), which doesn't hold its necessary limit points.