Prove that the bth projection map is continuous and open.

ForMyThunder

I am trying to prove that the bth projection map P_b:[tex]\Pi[/tex]X_a --> X_b is both continuous and open. I have already done the problem but I would like to check it.

1) Continuity:

Consider an open set U_b in X_b, then P_b^-1(U_b) is an element of the base for the Tychonoff topology on [tex]\Pi[/tex]X_a. Thus, P_b is continuous.

2) Openness:

Let U be an open set in [tex]\Pi[/tex]X_a and let p be a point in U. Then there exists a neighbourhood V of p contained in U. Thus, P_b(p) [tex]\in[/tex] P_b(V) and P_b(V) [tex]\subset[/tex] P_b(U) and every point P_b(p) of P_b(U) has a neighbourhood P_b(V). Thus, P_b(U) is open and P_b is an open map.

I'm less sure about whether my proof of openness is correct. Could you tell me if my answer is correct or not and where I need to improve in my proofs? This question came from General Topology by Stephen Willard (Section 8). Thank you in advance.

ForMyThunder

I just realized that my proof on openness is entirely wrong. Could anyone help me out on it?

rasmhop

You can't really know that [itex]P_b(V)[/itex] is open so this doesn't work. To show that a map is open it suffices to show that it maps basis elements to open sets so try considering an arbitrary basis element:
[tex]V = \prod U_a[/tex]
where the collection [itex]\{U_a\}[/tex] consists of open sets ([itex]U_a[/itex] open in [itex]X_a[/itex]). You should be able to show that [itex]P_b(V)[/itex] is open in [itex]X_b[/itex].

The reason that we only need to show it for basis elements is that once we know it's true for them we can take arbitrary unions of them to show that all open sets also map to open sets.

ForMyThunder

Yeah, I think i've figured it out now:

An arbitrary basis element is given by the finite intersection [tex]\bigcap[/tex]P_ai^-1(V_ai), where {a_i} is a finite subset of the index set and each V_ai is an open set in X_ai. So then, P_b([tex]\bigcap[/tex]P_ai^-1(V_ai) = [tex]\bigcap[/tex]P_b(P_ai^-1(V_ai)). When b[tex]\neq[/tex]a_i for all i, then this equals X_b. When b=a_i for some i, this equals V_ai. In either case, the image is an open set, thus P_b is an open map.

I wrote this out right after I saw that my first proof was wrong.