Quotient maps, open and closed maps
I am confused about the definition of a quotient map. If X and Y are topological spaces, then p:X-->Y is a quotient map if it is surjective and continuous and satisfies the following condition: A subset U of Y is open/closed in Y iff p-1(U) is open/closed in X.
I am a bit confused about how p can be a quotient map, but not an open/closed map. A map p is open/closed if for each open/closed subset A of X, p(A) is open/closed in Y. If we write the "iff" statement in the definition of quotient map as a biconidtional, isnt the definition of open/closed map simply the(<==) direction for the biconditional? Equivalently, isnt it the same as the "if" in the "iff". For if A is an open subset of X, then A is in dom(p) and hence there exists a subset U of Y such that A=p-1(U). So in the condition for an open map
(A open in X)==>( p(A) open in Y)
if we replace A with p-1(U) we get
(p-1(U) open in X)==>( p(p-1(U)) open in Y)
so
(p-1(U) open in X)==>( U open in Y)
which is the same as the statment that U is open in Y if p-1(U) is open in X and this is clearly satisfied when p is a quotient map.
Nonetheless, I have several examples in my book of quotient maps which are not open or which are not closed. Where is the error in the reasoning I layed out above?
p.s. I am thinking my mistake has something to do with assuming that p(p-1(U) = U I have a vague sense that is probably isnt correct
I am a bit confused about how p can be a quotient map, but not an open/closed map. A map p is open/closed if for each open/closed subset A of X, p(A) is open/closed in Y. If we write the "iff" statement in the definition of quotient map as a biconidtional, isnt the definition of open/closed map simply the(<==) direction for the biconditional? Equivalently, isnt it the same as the "if" in the "iff". For if A is an open subset of X, then A is in dom(p) and hence there exists a subset U of Y such that A=p-1(U). So in the condition for an open map
(A open in X)==>( p(A) open in Y)
if we replace A with p-1(U) we get
(p-1(U) open in X)==>( p(p-1(U)) open in Y)
so
(p-1(U) open in X)==>( U open in Y)
which is the same as the statment that U is open in Y if p-1(U) is open in X and this is clearly satisfied when p is a quotient map.
Nonetheless, I have several examples in my book of quotient maps which are not open or which are not closed. Where is the error in the reasoning I layed out above?
p.s. I am thinking my mistake has something to do with assuming that p(p-1(U) = U I have a vague sense that is probably isnt correct