The wealth gap should be closed by the poor working harder.
I have no qualifications outside of my GCSE's, I've worked hard, in crappy jobs, doing long hours, and, by the age of 30, I had two houses, one with a modest mortgage, one with a 75% mortgage, I had two cars, a motorbike, nice furniture, a big TV, all the consoles, and a holiday every year.
When I moved out from my my mum and dads at 18, I had no job, no house,nowhere to live, my motorbike (that was on finance), a rucksack full of clothes and about £500 in my current account.
I didnt claim dole, I didnt take handouts, (in tory Britain at that point as well 1994) I got a crappy warehouse job that paid £200 a week, got a flat, met my first wife and after a year we managed to save enough to put a deposit down on our first house (two bed terrace £32,500 brand new).
The poor should make something of themselves, unless you are so ill that you cannot move you should get nothing and have to support yourself.