Uncertain about premise for proof.

[madsmh]

I am reading §22 of Topology by Munkres, in Theorem 22.2 the function g is said to be constant on each set p^(-1)({y}). However the only explicit property given in Corollary 22.3 to the function g is that it is continuous and surjective, but Theorem 22.2 to g in the proof. Is it implied that g in 22.3 also has the properties given in Theorem 22.2?

.. Mads

 

[micromass]

Hi madsmh!

No, it isn't implicitly assumed that g or p has that property. However, it can be shown that p does have the correct properties of 22.2. Indeed, an arbitrary element of X* has the form [itex]g^{-1}(z)[/itex]. We must show that g is constant on sets of this form.
This is true since by definition x is in [itex]g^{-1}(z)[/itex] if g(x)=z. So all elements in [itex]g^{-1}(z)[/itex] are being sent to z. So g is constant on sets of the form [itex]g^{-1}(z)[/itex]. So the premises of 22.2 are satisfied.

 

[madsmh]

Thanks! :)