Find a Topological space where any sequence converges to any point. What the hell. Do you find the space worthless? I do. In Bengali we do say it "Ekkebare ja ta"
(এক্কেবারে যা তা).
But yes the first thing that strikes into your mind is that the topological space can not be Hausdorff. Because in Hausdorff space a sequence can have at most one limit. I found it as a exercise problem in Munkres. The topology was mentioned and Munkres asked to find a limit point of a specific sequence.
Consider any set with co-finite topology, that is in this topology any set, whose complement is open, is a open set. In Topological sense we say a sequence xn converges to x , if for any open ball containing x contains infinitely many points of the sequence. Let X be a space with co-finite topology. xn be any sequence in X and let y be any point in X. If we consider any open ball B containing y that contains all points of X with y except finitely many points. Now the sequence xn has also infinitely many points. Thus all but finitely many points of xn belongs to B. Hence y is a limit point of xn.
But yes the first thing that strikes into your mind is that the topological space can not be Hausdorff. Because in Hausdorff space a sequence can have at most one limit. I found it as a exercise problem in Munkres. The topology was mentioned and Munkres asked to find a limit point of a specific sequence.
Consider any set with co-finite topology, that is in this topology any set, whose complement is open, is a open set. In Topological sense we say a sequence xn converges to x , if for any open ball containing x contains infinitely many points of the sequence. Let X be a space with co-finite topology. xn be any sequence in X and let y be any point in X. If we consider any open ball B containing y that contains all points of X with y except finitely many points. Now the sequence xn has also infinitely many points. Thus all but finitely many points of xn belongs to B. Hence y is a limit point of xn.
