Hcfyk

2 $begingroup$ My question is: if you prove that something is true for all points in a topological space or a subset of some topological space, does that imply that this property holds for the whole topological space or the subset of the topological space? EDIT: more concrete if you have a topological space where all of its points are closed then is this space also closed? If that even makes sense. If this is true am I then allowed to pick an arbitrary point of the space and then show that since it holds for this one point then the topological space has this property? general-topology share | cite | improve this question edited yesterday