Let's analyze each of the options for whether they form an open set when considering the subset $A = (-\infty, 0]$ of the real line. Recall that an open set in the real line is a set where, for each of its points, there exists an interval around it that is entirely contained within the set. So, we need to check each of these sets accordingly.

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

• Here, $A = (-\infty, 0]$ and we are intersecting it with the interval $[0,1]$.
• $A \cap [0,1] = (-\infty, 0] \cap [0,1] = \{0\}$.
• The set $\{0\}$ is a single point, and single points are not open in the standard topology on the real line.

Thus, Option A is not open.

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

• Here, we are intersecting $A = (-\infty, 0]$ with the set $(-\infty,-1)$.
• $A \cap (-\infty,-1) = (-\infty, 0] \cap (-\infty,-1) = (-\infty, -1)$.
• The set $(-\infty, -1)$ is open in the standard topology on the real line because it contains an open interval for every point in the set.

Thus, Option B is open.

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

• Here, we are taking the union of $A = (-\infty, 0]$ with the point $\{ \frac{1}{2} \}$.
• The set $A \cup \{ \frac{1}{2} \} = (-\infty, 0] \cup \{ \frac{1}{2} \}$.
• The point $\frac{1}{2}$ is not part of $A$, and adding a single point to a set that is not open makes the union not open (since $\frac{1}{2}$ does not have a neighborhood around it fully contained within the set).

Thus, Option C is not open.

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

• Here, we are taking the union of $A = (-\infty, 0]$ with the interval $(-1,1)$.
• $A \cup (-1,1) = (-\infty, 0] \cup (-1,1) = (-\infty, 1)$.
• The set $(-\infty, 1)$ is open because it is an open interval.

Thus, Option D is open.

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

• Here, we are taking the union of $A = (-\infty, 0]$ with the interval $(0.1,1)$.
• $A \cup (0.1,1) = (-\infty, 0] \cup (0.1,1)$.
• This set is not open because the union has a "gap" at $0$, which is a boundary point and not an interior point, preventing the set from being open.

Thus, Option E is not open.

Summary of Results:

• Option A: Not open.
• Option B: Open.
• Option C: Not open.
• Option D: Open.
• Option E: Not open.

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Would you like a more detailed explanation on any specific point, or do you have any questions?

Here are 5 related questions for further exploration:

1. What is the definition of an open set in the real line?
2. Can a set that contains isolated points ever be open?
3. How do unions and intersections affect the openness of sets?
4. What is the difference between a closed set and an open set?
5. What role do boundary points play in determining whether a set is open?

Tip: To determine if a set is open, check if for each point in the set, there exists a neighborhood around it that lies entirely within the set