Open sourcing Android is arguably what made Android thrive and was amongst the smartest things that Google ever did, otherwise phone makers and careers would have continued preferring something they can have some control over, like Symbian. It's also arguably why Windows Phone is for the time being dead, with Android and iOS having a staggering 96% of the market, as nobody wants another tightly controlled platform/OS, as iOS itself fulfills that role pretty well and you can't beat Apple at its own game.
I don't know which definition you're referring to, but it's contrary to basic microeconomics. Open-source often fulfills one of these roles: (1) needed substitutes to prevent a dangerous competitor from completely controlling the market (e.g. Android vs iOS), (2) complementary products that increase the desirability of something else (e.g. Google Search), or (3) decreasing the cost of development (e.g. your favorite open-source programming language).
Also, the value of Gmail is not in its source-code and has nothing to do with it.
Open sourcing android was the only option they had as 1)Google support is non existent and 2)they didn't have the hardware or physical distribution at the time and 3)Apple showed the world how mobile operating systems should be done and making their OS closed would be playing a game of metoo which wouldn't let them stand out in any way. Also Google doesn't have any business motive to make profit off android because their whole business model is about selling user data.
if gmails source code is not its value what is it? I'm pretty sure gmails ability to search and functionality are what make it stand out which is easily replicable by taking the source code. If the source code isn't valuable, you're saying it's the brand that's valuable because without the actual product the only thing left is the gmail brand.
Gmail's value is the service it provides, which relies just as much on the infrastructure as the source code.
Back to your original statement - would you consider Linus dumb for open sourcing Linux? Open sourcing a product can be a very smart move - it gives certainty of supply, and encourages other enthusiasts to share the building of it.
Back in the day, silicon chip makers would allow their competitors to make copies of their chips so that consumers would know that if either company went bust they still could procure the chips they needed to build their products. If Google kept Roboto (and other elements of Android) closed, mobile phone makers would worry that if Google turned evil they could be left with hardware with no operating system.
And by open sourcing Roboto, the world's font enthusiasts get to play with it. Roboto has already evolved considerably within Google, and it's likely that this process will be accelerated with a wider set of participants.
So Google wins because it gets a better font for Android, and it's phone makers are happier to use Android. Seems like a smart move to me...
I don't know which definition you're referring to, but it's contrary to basic microeconomics. Open-source often fulfills one of these roles: (1) needed substitutes to prevent a dangerous competitor from completely controlling the market (e.g. Android vs iOS), (2) complementary products that increase the desirability of something else (e.g. Google Search), or (3) decreasing the cost of development (e.g. your favorite open-source programming language).
Also, the value of Gmail is not in its source-code and has nothing to do with it.