My rebbe o'h said that making an umbrella is like a folding chair.
Whom did he hold like? Of course, as you point out, that is only where there is an eruv chatzerot.
Ktiva vachatima tova
Rabbi Ed Goldstein
Assuming your Rebbe o.h. was citing a source as to why opening an umbrella is prohibitted, he must have been referring to Kisei Galin, about which Abaye rules 'Patur Aval Asur'.
But this is strange, since the two have nothing in common. Kisei Galin is Asur either because one is making an Ohel Arai (by setting up first the walls and then the Ohel, as Tos. explains), or because one may come to tighten the screws, like Rashi.
An umbrella on the other hand, has neither of those two specifications, and ought therefore to be permitted.
In similar vein, the Tiferes Yisrael (see final paragraph) rejects the proof to permit opening an umbrella from Kisei shel Perakim (opening a folding chair) which is permitted, since a folding chair does not serve as a shade, whereas an umbrella does.
One of the sources for prohibitting an umbrella is 'Si'ana', which according to the Gemara's conclusion, is a stiff wide-brimmed hat of at least a Tefach, which resembles an Ohel (according to Tosfos 138b DH 'Ha de'Mihadek'). And if anything, an umbrella is worse, since it is made in order to protect from rain (as we just explained), which the Poskim consider a Chumra regarding the Dinim of Ohel.
See also Tiferes Yisrael, in his introduction to Mishnayos Shabbos (Kupas ha'Rochlim), in K'lala de'Boneh Siman 3, who discusses this issue at length, and interestingly, concludes that opening an umbrella is permitted, though the Poskim do not agree with him.
Wishing you and yours a 'G'mar Chasimah Tovah'.
be'Virchas Kol Tuv
Eliezer Chrysler
1) The Chatam Sofer is misrepresented. He actually says that it is mutar both midorayta and mederabanan, and it is only a middat chasidut to be machmir.
2) MB does not say "that one is prohibited to carry it on Shabbos because wherever a person walks while he carries the umbrella, he makes a new Ohel in that place (since he is holding the umbrella for the sake of using it as an Ohel, which protects the space underneath it)." Rather he says that whenever you carry it or wherever you carry it, it is assur. It is not a problem of creating new ohels, and hotzaa is an extra problem.
Jonathan Ziring, New York, NY USA