Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient topology on Y defined to be: {V S Y|p(V) is open in X} is a topology.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify
that the quotient topology on Y defined to be: {V S Y|p (V) is open in X} is a
topology.
Definition:
If X is a space and A is a set and if p: X → A is a surjective map, then there exists
exactly one topology T on A relative to which p is a quotient map; it is called the
quotient topology induced by p.
Proof:
Let J = {V C Y\p¬'(V) is open in X}.
1) p-1(Y) = X is open in X.
Therefore, Y E JT.
p-²(Ø) = Ø is open in X.
Therefore, Ø E T.
2) Let U,V E T, then p-(U) and p-(V) are open in X.
Then, p-1(U) n p-'(V) is open in X.
Then, p-(UnV) = p-'(U) np-(V) is open in X.
Therefore, Un VET.
3) Let V, E T, where j€ J is an index set.
Then, p-1(V;) is open in X.
Then, p-(Uje, V;) = UjejP¯(V,).
Therefore, Ujej V, ET.
Therefore, the quotient topology on Y defined by T is a topology.
Transcribed Image Text:Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient topology on Y defined to be: {V S Y|p (V) is open in X} is a topology. Definition: If X is a space and A is a set and if p: X → A is a surjective map, then there exists exactly one topology T on A relative to which p is a quotient map; it is called the quotient topology induced by p. Proof: Let J = {V C Y\p¬'(V) is open in X}. 1) p-1(Y) = X is open in X. Therefore, Y E JT. p-²(Ø) = Ø is open in X. Therefore, Ø E T. 2) Let U,V E T, then p-(U) and p-(V) are open in X. Then, p-1(U) n p-'(V) is open in X. Then, p-(UnV) = p-'(U) np-(V) is open in X. Therefore, Un VET. 3) Let V, E T, where j€ J is an index set. Then, p-1(V;) is open in X. Then, p-(Uje, V;) = UjejP¯(V,). Therefore, Ujej V, ET. Therefore, the quotient topology on Y defined by T is a topology.
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