Is e an Accumulation Point in This Topology?

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Discussion Overview

The discussion revolves around the concept of accumulation points in a specific topological space defined by a set and a topology. Participants explore the definition of accumulation points, provide examples, and question the status of certain points within the topology.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant defines an accumulation point in a topological space and provides an example with the set X={a,b,c,d,e} and topology T={∅,{a,b},{b,c,d},{a,b,c,d},X}.
  • Another participant points out a missing open set {b} in the topology T and discusses the implications for identifying accumulation points.
  • It is noted that a, c, and d are accumulation points of {b,c,d}, while b is not considered an accumulation point.
  • There is uncertainty regarding whether e should be considered an accumulation point due to the absence of open sets containing e, leading to a discussion about the definition's antecedent.
  • One participant asserts that e is indeed an accumulation point since the open set X contains e and intersects with any subset.
  • A question arises about the terminology for points like e that are not contained in any open set other than X, with references to isolated points.
  • Another participant describes the topology as exhibiting pathological properties, such as every sequence converging and extreme compactness, suggesting that the space is exotic and does not resemble typical topological spaces.

Areas of Agreement / Disagreement

Participants generally agree on the definition of accumulation points and the identification of certain points as accumulation points. However, there is disagreement regarding the status of point e and the terminology used for points not contained in open sets other than X.

Contextual Notes

The discussion highlights limitations in the definition of the topology and the implications of missing open sets. The properties of the described space are noted to be unusual and may not align with standard topological intuitions.

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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Arian.D said:
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!
 
micromass said:
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?
 
Arian.D said:
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.
 

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