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Apr 1, 2016 at 6:47 comment added Dominic van der Zypen Thanks @Gro-Tsen - will think about it and post a new question in a week or so
Mar 31, 2016 at 15:06 comment added Gro-Tsen @DominicvanderZypen: I'll rather let you do that, but I suggest you give the matter further thought to decide what the best question should be, because it's not entirely clear. You probably want to require $C$ to be $(C,c_0,c_1)$-connected and $c_0$ and $c_1$ (hence any two points) to be connected by a path and sprinkle some magic "Hausdorff" dust in various places, but I'm not sure what else.
Mar 31, 2016 at 15:00 comment added Gro-Tsen @SimonHenry: There are still the indiscrete spaces, but you're right, it's even worse than I thought.
Mar 31, 2016 at 14:59 comment added Dominic van der Zypen @Gro-Tsen very good point, can you ask the question with your additional requirement?
Mar 31, 2016 at 14:49 vote accept Dominic van der Zypen
Mar 31, 2016 at 14:41 comment added Simon Henry With regard to your last remarks: In fact it would already be nice to have an example where $C$-connectedness is a non trivial notion with (i.e. there is as least one non trivial space that is $C$-connected)
Mar 31, 2016 at 12:48 history answered Gro-Tsen CC BY-SA 3.0