I came here to post about my cat doing this and found your thread instead
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This is incredibly long and if you want to get to the point, skip to the bottom.*
I walk him round the block in daylight (city streets) for up to 30 minutes a day unless it's cold or wet - that's his choice to stay in, not mine. No lead. He walks alongside me or in front on the pavement. Sometimes trots. He's never been taught to do that.
Each house has a basement and he explores them for a few minutes and I've noticed he has favourites and completely ignores some others. He comes when I call his name mostly and pays more attention if I rattle some treats.
He can't resist an open door or doors that have brass kickplates - he stares at his reflection. I don't let him go into open doors in case there's a dog or people who don't like cats.
He was my mum's cat and I got him when she died nearly two years ago. He was 11 and had been an exclusively indoor cat with his brother since a kitten. His brother died a few years ago.
In the first spring after she died I let him out into my back garden. It's reasonably secure with 5ft walls. He could climb that - I found that out at the weekend when he jumped over 4ft from a standing start with ease (did I say he was 13?) - but preferred to sunbathe for hours.
I wouldn't let him out alone because I live between two extremely busy roads. He came with me into the front basement when I cleaned the windows or swept up. The walls are about 12ft but there are steps and a gate he could get through but he was more interested in mooching around a bin store under the steps which smelled excitingly of who knows what?
But the front door click had a siren's call because it was forbidden so that's why I started walking him this spring.
I've spoken to more of my neighbours in six months than in the 23 years I've lived here. Everyone loves him. I was hoping for a slightly better image than: 'Oh, you're the cat lady' but that's my lot.
All but two dog walkers round here keep their dogs on leads. The two that don't have friendly dogs. I scoop him up but if I let him out alone, he'd meet them anyway.
As it happens he has no fear, no road sense and thinks everyone is his friend. He wouldn't last five minutes on his own. Everyone so far has been his friend. Loads of people know his name. They don't ever ask mine 
- Which brings me to the point of this shaggy dog tale. We were walking on Monday and I was talking to a woman who was making a big fuss of him when another woman with her small grand daughter butted in.
Her grand daughter was dying to touch him. I said he was friendly and she could. He's a cat though. Unaccompanied cats are more common than dogs so if you were scared of meeting animals, cats would be more of a problem than dogs.
The woman kept her grand daughter away, fair enough. Through she did behave like he was a tiger and one with mange, at that. I'm sure Nobby didn't want to be grabbed either, though she looked like a nice little girl.
Then the woman asked why I was walking him and I said I couldn't trust him out on his own because he's so friendly and so street and traffic clueless after being an indoor cat for 11 years.
Then she said: 'Cats can go out on their own and if they get run over that just happens. I don't understand why you're doing this. It's just a cat.' 
WTF has it got to do with her that I spend 30 minutes a day eccentrically trailing after my cat and appearing like a burglar's look out while he mooches round basements?
Thanks if you got this far 