# Topology/Sequences

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.

## Convergence[edit]

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

## Exercises[edit]

- Give a rigorous description of the following sequences of natural numbers:

(i)

(ii) - 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