Accumulation point definition.

[Arian.D]

We say a point x in X (which is a topological space) is an accumulation point of A if and only if any open set containing x has a non-empty intersection with A-{x}.

Well, I'm creating examples for myself to understand the definition.
Suppose X={a,b,c,d,e} and define T={∅,{a,b},{b,c,d},{a,b,c,d},X}. T is a topology on X. Now I'm trying to find the set of all accumulation points of {b,c,d}.

a,c and d are accumulation points of {b,c,d}, b is not an accumulation point of it, but I'm not sure if I should consider e an accumulation point of {b,c,d} or not because there is no open set containing e in my topology defined on X. Should I consider e an accumulation point because the antecedent in the definition (where it assumes that there exists an open set containing that point) is false for e?

 

[micromass]

We say a point x in X (which is a topological space) is an accumulation point of A if and only if any open set containing x has a non-empty intersection with A-{x}.

Well, I'm creating examples for myself to understand the definition.
Suppose X={a,b,c,d,e} and define T={∅,{a,b},{b,c,d},{a,b,c,d},X}. T is a topology on X. Now I'm trying to find the set of all accumulation points of {b,c,d}.

There is something missing in the definition of T. You're missing {b}.

a,c and d are accumulation points of {b,c,d}, b is not an accumulation point of it, but I'm not sure if I should consider e an accumulation point of {b,c,d} or not because there is no open set containing e in my topology defined on X. Should I consider e an accumulation point because the antecedent in the definition (where it assumes that there exists an open set containing that point) is false for e?

There is an open set containing e: the set X is open and contains e!

 

[Arian.D]

There is something missing in the definition of T. You're missing {b}.

Oops, yea.

There is an open set containing e: the set X is open and contains e!

And X has a non-empty intersection with anyone of its subsets. Good! so it's an accumulation point. Thanks.

One more thing, What do we call a point like e that is not contained in any open set in the topology excluding X? If {a} is in the topology we call a isolated, right? Do we call e by a particular name in topology?

 

[micromass]

Oops, yea.



And X has a non-empty intersection with anyone of its subsets. Good! so it's an accumulation point. Thanks.

One more thing, What do we call a point like e that is not contained in any open set in the topology excluding X? If {a} is in the topology we call a isolated, right? Do we call e by a particular name in topology?

I'm not aware of any specific name. But the situation you describe is very pathological. The space exhibits some very weird properties such as

- Every sequence (and even filter and net) converges.
- The space is extremely compact: every open cover has {X} as subcover.

In fact, the previous two properties are equivalent and imply the existence of a point a whose only neighborhood is X.

So the space you describe is quite exotic (and interesting!), but it does not ressemble at all the nice spaces we expect in topology.

 

[blue_raver22]

I would say that it depends on the context and the purpose of considering accumulation points in this specific topology. If we are only interested in accumulation points that can be found in open sets, then we can say that e is not an accumulation point of {b,c,d}. However, if we want to consider all possible accumulation points, including those that may not have an open set containing them, then we can say that e is an accumulation point of {b,c,d}. It is important to be clear about the criteria for determining accumulation points in a specific topology to avoid confusion and ensure accurate results in our research.