Prove
Let f : X ? Y be continuous. Then there is a quotient map q of X onto a space Z and a one-one...

Question

Prove
Let f : X ? Y be continuous. Then there is a quotient map q of X onto a space Z and a one-one continuous map h of Z into Y such that f=h ?q.

Robert · Accepted Answer

If Z is any quotient space of X then you must agree that we ca not find an
injective map h : Z → Y . In particular we can take Z = X, that is with trivial
relation, then q = Identity and there is no map h : Z → Y such that h ◦ q = f .
Because if h ◦ q = f then as q = Identity we have h = f but f may not be
injective. Hence for any quotient space Z of X, this assertion is not true.

Prove Let f : X ? Y be continuous. Then there is a quotient map q of X onto a space Z and a one-one continuous map h of Z into Y such that f=h ? q.

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