什么样的商映射既不是开映射也不是闭映射？我觉得商映射既然是连续的，那么要么是开得要么是闭的，谁能帮忙举个例子？
还有如何用商映射得到环面，Clain瓶，射影平面，射影空间？其需要的等价条件是什么？
谢谢了先

1. Let I=[-1,1], X={0,1}, f the characteristic function of [0,1] in I, endow with X the quotient topology, i.e. the topology {empty set, {0}, X}. It is easy to verify that f is a quetient map, but is neither open nor closed.

2. A quetient map f:X->Y is open(closed) provided that for each open(closed) subset B of X f^-1(f(B)) is also open(closed) in X.

3. I as above.
　　　　I^2/~ is the torus, here ~ identifies (-1,y) with (1,y), and (x,-1) with (x,1).
　　　　I^2/~ is the Klein bottle, here ~ identifies (-1,y) with (1,y), and (-x,-1) with (x,1).
　　　　bd(I^n+1)/~ is the projective space of dimension n, here ~ identifies x with -x where x∈bd(I^n+1). If n=2, we get the projective plane.