Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following condition holds: q(E) is closed if and only if E is closed. Proof: First, suppose that q is a quotient map. Then, V is open in Y if and only if q¬'(V) is open in X. Now, E is closed if and only if E° is open if and only if q-'(E°) is open. q(E°) = [q(E)]° Therefore, q is onto. E is closed if and only if [q-(E)]° is open. Let E be closed if and only if q-(E) is closed. If V is open then Ve is closed. Then, q-(VC) is closed and [q-1(V)]° is also closed. Therefore, q-1(V) is open. Hence, q is a quotient map.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the
following condition holds: q1(E) is closed if and only if E is closed.
Proof:
First, suppose that q is a quotient map. Then, V is open in Y if and only if q¬'(V) is
open in X.
Now, E is closed if and only if E° is open if and only if q-'(E°) is open.
q-'(E°) = [q¬(E)]°
Therefore, q is onto.
E is closed if and only if [q-(E)]° is open.
Let E be closed if and only if q-(E) is closed.
If V is open then Ve is closed.
Then, q-(VC) is closed and [q-(V)]° is also closed.
Therefore, q-1(V) is open.
Hence, q is a quotient map.
Transcribed Image Text:Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following condition holds: q1(E) is closed if and only if E is closed. Proof: First, suppose that q is a quotient map. Then, V is open in Y if and only if q¬'(V) is open in X. Now, E is closed if and only if E° is open if and only if q-'(E°) is open. q-'(E°) = [q¬(E)]° Therefore, q is onto. E is closed if and only if [q-(E)]° is open. Let E be closed if and only if q-(E) is closed. If V is open then Ve is closed. Then, q-(VC) is closed and [q-(V)]° is also closed. Therefore, q-1(V) is open. Hence, q is a quotient map.
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