The Closure of Set A in the Standard Topology on R
Let (R) be the set of real numbers with the standard topology. The closure of a set (A) in (R), denoted as (\overline{A}), is the smallest closed set containing (A).
For the set (A = [45, 46)), the closure is (\overline{A} = [45, 46]). This is because:
- (\overline{A}) must contain (A).
- (\overline{A}) must be closed.
- Any closed set containing (A) must also contain (\overline{A}).
Since ([45, 46]) is the smallest closed set containing (A), it is the closure of (A).
Related Questions and Answers
- What is the closure of the set ((0, 1)) in the standard topology on (R)? > Answer: ([0, 1])
- Is the set ({π}) closed in the standard topology on (R)? > Answer: Yes
- What is a neighborhood of a point (x) in the standard topology on (R)? > Answer: An interval containing (x)
- Is the set (\mathbb{Q}) open in the standard topology on (R)? > Answer: No
- What is the boundary of the set ((-\infty, 0])? > Answer: (-1)
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