Accumulation point definition.

  • Thread starter Arian.D
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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?
 

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  • #2
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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!!
 
  • #3
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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 any one 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?
 
  • #4
22,129
3,297
Oops, yea.



And X has a non-empty intersection with any one 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.
 

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