North Center has become one of the most competitive places to buy a home on Chicago’s North Side, and the surge is still in its early stages. For years, people moved here for the schools and stayed for the quiet blocks. Now the neighborhood is rebuilding its commercial core around them. More than 300 new homes, plus new shops, bars and restaurants, are moving through approvals and permits, and almost none of it has opened yet. The demand is already here. The neighborhood people are buying into is still being built.