Discussion
Consider the metric spaces \(X = (\mathbb{R}, d_{1})\) and \(Y = ([0,1], d_{2})\) with the metrics defined by \(d_{1}(x,y) = |x - y|,\ x, y \in \mathbb{R}\) and \(d_{2}(x,y) = |x - y|,\ x, y \in [0,1]\), respectively. Then, which one of the following is TRUE?
Answer: B
\([0, \tfrac{1}{4})\) is open in Y but not in X.
In \(X = \mathbb{R}\), no open interval around 0 lies inside \([0, \tfrac{1}{4})\), because every such interval contains negative numbers. So the set is not open in X.
In Y = [0, 1] with the same metric, the open ball of radius 1/4 about 0 is \(\{y \in [0, 1] : |y| \lt \tfrac{1}{4}\} = [0, \tfrac{1}{4})\), and every other point of the set has a small ball inside it. So the set is open in Y; it equals \((-\tfrac{1}{4}, \tfrac{1}{4}) \cap [0, 1]\), open in the subspace topology. Openness depends on the ambient space, which is the point of the question.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development