Exercise · ZW-OPEN-SET-IN-A-METRIC-SPACE-007 · ★★☆ Intermediate
What is \(\bigcap_{n\ge1}\left(-\tfrac1n,\tfrac1n\right)\) in \(\mathbb{R}\)?
Choose one option.
Show answer
- D. It is \(\{0\}\), not open
Explanation
Only \(0\) lies in every interval. No ball around \(0\) fits inside \(\{0\}\) under the usual distance, so it is not open: an infinite intersection of open sets need not be open.
Review the concept: Open set in a metric space
A subset \(U\) of a metric space \((X,d)\) is open if every point \(x\in U\) is the center of some open ball \(B(x,r)\) contained in \(U\).
\[\forall x\in U\ \ \exists r>0:\ B(x,r)\subseteq U\]
More practice problems: open set in a metric space
- ZW-OPEN-SET-IN-A-METRIC-SPACE-001 In usual ℝ, which of these sets is open?
- ZW-OPEN-SET-IN-A-METRIC-SPACE-002 Let U=(0,3) in ℝ. What is the largest radius r with B(1,r)⊆ U?
- ZW-OPEN-SET-IN-A-METRIC-SPACE-003 Why is (0,1] not open in usual ℝ?
- ZW-OPEN-SET-IN-A-METRIC-SPACE-004 In ℝ^2 with the Euclidean distance, which of these sets is open?
- ZW-OPEN-SET-IN-A-METRIC-SPACE-005 In a metric space (X,d) with the discrete metric, which subsets are open?
- ZW-OPEN-SET-IN-A-METRIC-SPACE-006 Let U=\x∈ℝ: x^2<4\ with the usual distance. Is it open?
- ZW-OPEN-SET-IN-A-METRIC-SPACE-008 Let U=B(0,1) in ℝ. What is the largest radius r with B(0.9,r)⊆ U?
- ZW-OPEN-SET-IN-A-METRIC-SPACE-009 Is ℚ an open subset of usual ℝ?
← ZW-OPEN-SET-IN-A-METRIC-SPACE-006 ZW-OPEN-SET-IN-A-METRIC-SPACE-008 →