I finally understood, this is the same technique you used to proove that l^inf in inseparable. The argument I couldn't understand back then is that because S is dense in X then so is dense in C, but since all open balls in C are disjoint, S must have at least the same cardinality as C, C is an uncountable infinite set, so S in not countable, therefore X is inseparable. The catch is that you have to define such a set as C in X and that's tricky, for this to work out C must be a discrete metric space in order for your balls to be disjoint. Brilliant! Thank you very much sir.
Hi Leandro, could you tell me if my argument is correct?. Let S be a countable dense set in X. Then we take some function f which belongs to the set C mentioned in the video(set C being the set of functions which are 1 at one point and zero elsewhere). 0
@@ey3796 hi there, it's been a year since this comment haha I should revisit these concepts in order to respond to you. I've been studying other branches of math and well, I've gotten rusty hahaha
So, you could generalize this, there's nothing special about B[a,b] or l^inf. Let X be a metric space, if C is an uncountable subset of X whith a discrete metric, then any dense subset of X must be uncountable, so X is inseparable.
I finally understood, this is the same technique you used to proove that l^inf in inseparable. The argument I couldn't understand back then is that because S is dense in X then so is dense in C, but since all open balls in C are disjoint, S must have at least the same cardinality as C, C is an uncountable infinite set, so S in not countable, therefore X is inseparable. The catch is that you have to define such a set as C in X and that's tricky, for this to work out C must be a discrete metric space in order for your balls to be disjoint. Brilliant! Thank you very much sir.
Hi Leandro, could you tell me if my argument is correct?. Let S be a countable dense set in X. Then we take some function f which belongs to the set C mentioned in the video(set C being the set of functions which are 1 at one point and zero elsewhere). 0
@@ey3796 hi there, it's been a year since this comment haha I should revisit these concepts in order to respond to you. I've been studying other branches of math and well, I've gotten rusty hahaha
So, you could generalize this, there's nothing special about B[a,b] or l^inf.
Let X be a metric space, if C is an uncountable subset of X whith a discrete metric, then any dense subset of X must be uncountable, so X is inseparable.