Projection and quotient map

1. Nov 27, 2008

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).

2. Nov 27, 2008

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.

Last edited: Nov 27, 2008