Topology/Sequences

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Topology
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A sequence in a space X is defined as a function from the set of natural numbers into that space, that is f:\mathbb{N}\to X. The members of the domain of the sequence are f(1),f(2),\ldots and are denoted by f(n)=a_n. The sequence itself, or more specifically its domain are often denoted by \left\langle a_i\right\rangle.

The idea is that you have an infinite list of elements from the space; the first element of the sequence is f(1), the next is f(2), etc. For example, consider the sequence in \mathbb{R} given by f(n) = 1/n. This is simply the points 1,1/2,1/3,1/4,... Also, consider the constant sequence f(n) = 1. You can think of this as the number 1, repeated over and over.


Convergence[edit]

Let X be a set and let \mathcal{T} be a topology on X
Let \left\langle x_i\right\rangle be a sequence in X and let x\in X

We say that "\left\langle x_i\right\rangle converges to x" if for any neighborhood U of x, there exists N\in\mathbb{N} such that n\in\mathbb{N} and n>N together imply x_n\in U

This is written as \lim_{n\to\infty}x_n=x


Exercises[edit]

  1. Give a rigorous description of the following sequences of natural numbers:
    (i) 1,2,3,4,5\dots
    (ii) 2,-4,6,-8,10,\ldots
  2. Let X be a set and let \mathcal{T} be a topology over X. Let x\in X and let U be a neighbourhood of x.
    Let U_1\subset U and x\in U_1. Similarly construct neighbourhoods U_i\subset U_{i-1} with x\in U_i\forall i. Let \left\langle x_i\right\rangle be a sequence such that each x_i\in U_i.

    Prove that \lim_{n\to\infty}x_n=x


Topology
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