Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Accumulation point definition.

  1. May 11, 2012 #1
    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?
  2. jcsd
  3. May 11, 2012 #2
    There is something missing in the definition of T. You're missing {b}.

    There is an open set containing e: the set X is open and contains e!!
  4. May 11, 2012 #3
    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?
  5. May 11, 2012 #4
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook