To determine which of the sets from the list are open, we must analyze each one carefully. Recall that a subset $S$ of the real line $\mathbb{R}$ is open if for every point $x \in S$, there exists an $\epsilon > 0$ such that the interval $(x - \epsilon, x + \epsilon) \subseteq S$.

We are working with the set $A = (-\infty, 0]$, which is a closed set because it contains all points up to 0, including 0. Now, let's evaluate each option:

A) $A \cap [0,1]$:

• This is the intersection of $(-\infty, 0]$ and $[0,1]$, which results in $\{0\}$ (the only point in both sets).
• A set containing a single point (here $\{0\}$) is not open because there is no interval around 0 that stays entirely within the set.
• Not open.

B) $A \cap (-\infty, -1)$:

• This is the intersection of $(-\infty, 0]$ and $(-\infty, -1)$, which gives $(-\infty, -1)$, an open interval.
• Open intervals are open sets in the real line, so this set is open.
• Open.

C) $A \cup \{ \frac{1}{2} \}$:

• This is the union of $(-\infty, 0]$ and $\{ \frac{1}{2} \}$. Since $(-\infty, 0]$ is closed and adding a single point $\frac{1}{2}$ does not make it open, the resulting set will not be open.
• Not open.

D) $A \cup (-1,1)$:

• This is the union of $(-\infty, 0]$ and the open interval $(-1,1)$.
• This union results in $(-\infty, 1)$, which is an open interval.
• Open.

E) $A \cup (0.1, 1)$:

• This is the union of $(-\infty, 0]$ and the open interval $(0.1, 1)$.
• Since the interval $(0.1, 1)$ does not affect the part of $A$, the set $A \cup (0.1, 1)$ is still closed to the left of 0 and thus not open.
• Not open.

Correct answers:

• B) $A \cap (-\infty, -1)$
• D) $A \cup (-1,1)$

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the differences between open and closed sets in topology?
2. Can a set be both open and closed (clopen) in the real line?
3. How does the definition of an open set change in higher dimensions (e.g., in $\mathbb{R}^2$)?
4. How does the concept of boundary points relate to open sets?
5. What is the interior of a set, and how is it determined?

Tip: In general, open intervals $(a, b)$ in the real line are always open sets, while closed intervals $[a, b]$ are closed sets.