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Any composition of quotient maps is a quotient map. An injective quotient map is a homeomorphism. Ifq: X→Y is a quotient map, a subset KCY is closed if and only if q-¹(K) is closed in X.

Any composition of quotient maps is a quotient map. An injective quotient map is a homeomorphism. Ifq: X→Y is a quotient map, a subset KCY is closed if and only if q-¹(K) is closed in X.

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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prove each proposition for a,b, and c

**Proposition 3.62 (Properties of Quotient Maps).** (a) Any composition of quotient maps is a quotient map. (b) An injective quotient map is a homeomorphism. (c) If \( q : X \rightarrow Y \) is a quotient map, a subset \( K \subseteq Y \) is closed if and only if \( q^{-1}(K) \) is closed in \( X \). --- ► **Exercise 3.63.** Prove Proposition 3.62.
Transcribed Image Text:**Proposition 3.62 (Properties of Quotient Maps).** (a) Any composition of quotient maps is a quotient map. (b) An injective quotient map is a homeomorphism. (c) If \( q : X \rightarrow Y \) is a quotient map, a subset \( K \subseteq Y \) is closed if and only if \( q^{-1}(K) \) is closed in \( X \). --- ► **Exercise 3.63.** Prove Proposition 3.62.
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