Exercise · ZW-OPEN-SET-IN-A-METRIC-SPACE-010 · ★★★ Advanced
Let \(X=(0,1]\) with the usual distance. Is \((1/2,1]\) open in the space \(X\)?
Choose one option.
Show answer
- B. Yes, because in \(X\) we have \(B(1,1/2)=(1/2,1]\)
Explanation
Inside \(X\), \(B(1,1/2)=\{y\in(0,1]: |y-1|<1/2\}=(1/2,1]\), and the other points have small balls inside. Openness depends on the ambient space \(X\), not on \(\mathbb{R}\).
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-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?