Topology/Sequences

Topology
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A sequence in a space is defined as a function from the set of natural numbers into that space, that is . The members of the domain of the sequence are and are denoted by . The sequence itself, or more specifically its domain are often denoted by .

The idea is that you have an infinite list of elements from the space; the first element of the sequence is , the next is , etc. For example, consider the sequence in given by . This is simply the points Also, consider the constant sequence . You can think of this as the number 1, repeated over and over.


ConvergenceEdit

Let   be a set and let   be a topology on  
Let   be a sequence in   and let  

We say that "  converges to  " if for any neighborhood   of  , there exists   such that   and   together imply  

This is written as  


ExercisesEdit

  1. Give a rigorous description of the following sequences of natural numbers:
    (i)  
    (ii)  
  2. Let   be a set and let   be a topology over  . Let   and let   be a neighbourhood of  .
    Let   and  . Similarly construct neighbourhoods   with  . Let   be a sequence such that each  .

    Prove that  


Topology
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