I am trying to prove that the bth projection map P(adsbygoogle = window.adsbygoogle || []).push({}); _{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.

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# Prove that the bth projection map is continuous and open.

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