Exercise · ZW-OPEN-SET-IN-A-METRIC-SPACE-004 · ★☆☆ Basic
In \(\mathbb{R}^2\) with the Euclidean distance, which of these sets is open?
Choose one option.
Show answer
- D. \(\{(x,y): x^2+y^2<4\}\)
Explanation
It is the open ball centered at the origin with radius \(2\), and every open ball is an open set. The others contain boundary points, where no ball fits inside.
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-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-007 What is bigcap_n≥1(-tfrac1n,tfrac1n) in ℝ?
- 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?
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