Is Every Projection Map a Quotient Map in Topology?

[math8]

Is a projection a quotient map?

I think a quotient map is an onto map p:X-->Y (where X and Y are topological spaces) such that
U is open/closed in Y iff (p)-1(U) is open/closed in X.

And a projection is a map f:X-->X/~ defined by f(x)=[x] where [x] is the equivalent class (for a relation ~) containing x.

I guess a projection is onto because for every equivalent class [x], there is an x that maps to it, but I am wondering if every projection is a quotient map (or if every projection is continuous).

 

[Unco]

Hi Math8,

In order to talk about whether your projection map is continuous, you need to have defined what the open sets in X/~ are. The standard procedure is to define the topology on X/~ to be the so-called quotient topology induced by f. That is, so that f is a quotient map; i.e., a set U in X/~ is open iff f^(-1)(U) is open in X.

Of course, one can invent a topology on X/~ so that f is not continuous. Say, partition the plane X=R2 into a closed half-plane and an open half-plane. Then X/~ has two points X1 and X2, where $$f^{-1}(X_1)$$ is a closed-half-plane, and $$f^{-1}(X_2)$$ is an open half-plane. Define the topology on X/~ to be the discrete one, so X1 and X2 are open, and f is not continuous as $$f^{-1}(X_1)$$ is not open in R2.