To demonstrate that the intervals $(a, \infty)$ and $(-\infty, a)$ are open sets, and that $[b, \infty)$ and $(-\infty, b]$ are closed sets, we rely on the definitions of open sets and closed sets in topology.

Open Sets

A subset $S$ of a metric space (such as $\mathbb{R}$ with the standard metric) is open if, for every point $x \in S$, there exists an $\epsilon > 0$ such that the open interval $(x - \epsilon, x + \epsilon)$ is entirely contained in $S$.

1. The Interval $(a, \infty)$:

Let $x \in (a, \infty)$. Then $x > a$. Choose $\epsilon = x - a$, which is positive because $x > a$. The open interval $(x - \epsilon, x + \epsilon) = (a, x + \epsilon)$ lies entirely in $(a, \infty)$. Thus, $(a, \infty)$ is open.

2. The Interval $(-\infty, a)$:

Let $x \in (-\infty, a)$. Then $x < a$. Choose $\epsilon = a - x$, which is positive because $x < a$. The open interval $(x - \epsilon, x + \epsilon) = (x - \epsilon, a)$ lies entirely in $(-\infty, a)$. Thus, $(-\infty, a)$ is open.

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Closed Sets

A subset $S$ of a metric space is closed if its complement $S^c$ is open, or equivalently, if $S$ contains all its limit points.

3. The Interval $[b, \infty)$:

The complement of $[b, \infty)$ is $(-\infty, b)$, which we have already shown to be an open set. Therefore, $[b, \infty)$ is closed. Additionally, any sequence in $[b, \infty)$ that converges will converge to a point in $[b, \infty)$, confirming it is closed.

4. The Interval $(-\infty, b]$:

The complement of $(-\infty, b]$ is $(b, \infty)$, which we have already shown to be an open set. Therefore, $(-\infty, b]$ is closed. Furthermore, any sequence in $(-\infty, b]$ that converges will converge to a point in $(-\infty, b]$, confirming it is closed.

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Conclusion

• Open sets: $(a, \infty)$, $(-\infty, a)$
• Closed sets: $[b, \infty)$, $(-\infty, b]$

Would you like me to go deeper into any part of the explanation or cover related topics?

Related Questions:

1. What is the difference between open and closed sets in topology?
2. Can a set be both open and closed in $\mathbb{R}$?
3. How are limit points related to closed sets?
4. How do half-open intervals fit into the classification of open/closed sets?
5. What are examples of neither open nor closed sets in $\mathbb{R}$?

Tip: In topology, the notions of open and closed sets depend on the space under consideration, so always verify the context.