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6. Let X be the space of real numbers with the usual topology, and let Q be the subspace of rational numbers. Let A =.{re Q: 0 < r < 1}, B = {re Q: 0 < r ≤ √2}, C = {re Q:-√2 ≤r ≤ √2). For each of the sets A, B, and C, tell whether in the subspace topology it is open, closed, or neither.

6. Let X be the space of real numbers with the usual topology, and let Q be the subspace of rational numbers. Let A =.{re Q: 0 < r < 1}, B = {re Q: 0 < r ≤ √2}, C = {re Q:-√2 ≤r ≤ √2). For each of the sets A, B, and C, tell whether in the subspace topology it is open, closed, or neither.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Topology: Please follow structured example in photo to solve Q26
26. Let X be the space of real numbers with the usual topology, and let Q be the subspace of rational numbers. Let A = {r EQ: 0 < r < 1}, B = {re Q:0 < r ≤ √2}, C = {re Q:-√2 ≤ r ≤ √2}. For each of the sets A, B, and C, tell whether in the subspace topology it is open, closed, or neither.
Transcribed Image Text:26. Let X be the space of real numbers with the usual topology, and let Q be the subspace of rational numbers. Let A = {r EQ: 0 < r < 1}, B = {re Q:0 < r ≤ √2}, C = {re Q:-√2 ≤ r ≤ √2}. For each of the sets A, B, and C, tell whether in the subspace topology it is open, closed, or neither.
s.bbcollab.com/collab/ui/session/playback ex Let (X,T) = (IR, usual top.) Let A = Q = set of rationals, Then (A, TA)=(Q, To) is a Subspace. Let S={PEQ: 0≤r≤1} 1/45+ I Is S open in (A, TA)? Is S closed in (A,TA)? fact in (Rusual), C(x5)= (x-r, x+r) fact PES every base el. cont p meets S claim cl₂S = [0, 1] S 117 Ę P& ²1 (5) (p<0) C(pi-p) = (p-(-p), p+(-p)) = (2p, 0) neet S=> does not → PACI S.
Transcribed Image Text:s.bbcollab.com/collab/ui/session/playback ex Let (X,T) = (IR, usual top.) Let A = Q = set of rationals, Then (A, TA)=(Q, To) is a Subspace. Let S={PEQ: 0≤r≤1} 1/45+ I Is S open in (A, TA)? Is S closed in (A,TA)? fact in (Rusual), C(x5)= (x-r, x+r) fact PES every base el. cont p meets S claim cl₂S = [0, 1] S 117 Ę P& ²1 (5) (p<0) C(pi-p) = (p-(-p), p+(-p)) = (2p, 0) neet S=> does not → PACI S.
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