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I am currently learning about limit points in my Topology class and am a bit confused. Going by this:

As another example, let X = {a,b,c,d,e} with topology T = {empty set, {a}, {c,d}, {a,c,d}, {b,c,d,e}, X}. Let A = {a,b,c} then a is not a limit point of A, because the open set {a} containing a does not contain any other point of A different from a. b is a limit point of A, because the open sets {b,c,d,e} and X containing b also contain a point of A different from b. Similarly, d and e are also limit points of A. This illustration suggests that a set can have more than one limit point.

What I do not understand is that {a,c,d} contains points different from a . Same goes for point c, {b,c,d,e} contains points other than c that are also in A but it is not a limit point? Can anyone explain this to me at all?