Suppose q: X → Y is a quotient map and that it is one-to-one. Show that q is a homeomorphism. (Hint: Since q is injective, for any A C X,q¬(q(A)) = A)).
Suppose q: X → Y is a quotient map and that it is one-to-one. Show that q is a homeomorphism. (Hint: Since q is injective, for any A C X,q¬(q(A)) = A)).
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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Can you please look over my proof and make suggestions/corrections?
Thank you,
Transcribed Image Text:Suppose q: X
→ Y is a quotient map and that it is one-to-one. Show that q is a
homeomorphism. (Hint: Since q is injective, for any A C X,q¬(q(A)) = A)).
Proof:
Suppose q: X → Y is a quotient map and that it is one-to-one.
Then, q is onto.
V is open in Y if and only if q-(V) is open in X.
Therefore, q is continuous.
If q-1: Y → X, then let A be an open subset of X.
Then, q(A) is a subset of Y.
q(q(A)) = A for all A C X.
Then, q(A) is open in Y.
Therefore, q-1 is continuous.
Therefore, q is a homeomorphism.
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