America likes to think of itself as a country where the doors to college are open to all, where anyone can earn a bachelor’s degree through hard work and academic merit, and where public support for higher education is unwavering. Reality belies the vision. The doors are wide open for the privileged, but opportunity narrows considerably for the less fortunate. The bachelor’s is eminently reachable for those who stick with their studies — indeed, the master’s degree is coming into vogue for many — but huge numbers of promising students who start college don’t finish. What about public support? The synergy between federal and state governments that propelled the phenomenal expansion of higher education in the 19th and 20th centuries has all but vanished, Suzanne Mettler writes in her provocative new book, “Degrees of Inequality.” The Cornell University political scientist argues that this drift in public policy over the past quarter-century — in Washington and in statehouses — has exacerbated gaps between the haves and the have-nots, undermining the ideal of college as an engine of upward social mobility.