# Topology/Sequences

 Topology ← Points in Sets Sequences Subspaces →

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

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

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 ← Points in Sets Sequences Subspaces →