# Topology/Lesson 4

## Introduction to Limits[edit | edit source]

This lesson will introduce the notion of a limit.

### Definition (sequence)[edit | edit source]

A *sequence* on a topological space is a function . Alternatively, it is a list where for all . The sequence is often denoted as

### Definition (limit of a sequence)[edit | edit source]

The point is a *limit* of the sequence if for every neighborhood , there is such that for all . In this case, we write and say that *converges* to .

Note that a sequence might have multiple limits. For example, in any space with the indiscrete topology, every sequence converges to every point of the space!

### Example[edit | edit source]

Let be the set of integers with the topology where is open if (called the finite complement topology). Let . Then we see that for any . Indeed, note that given any neighborhood , contains all but finitely many points of . Let be the maximum of all of the numbers not contained in . Then for all , we see that , hence .

### Theorem[edit | edit source]

If a space is Hausdorff and the sequence in has a limit, then that limit is unique.

The proof of this theorem is left as an exercise to the student. The hint is to assume that there are two distinct limits and show that this leads to a contradiction.

### Theorem[edit | edit source]

If is a continuous function and in then in Y.

#### Proof[edit | edit source]

Let be an open neighborhood of . Since is continuous, is open in and, by definition, contains . Therefore, there is such that for all . Therefore, . Thus, .

The converse of this theorem is, in general, false. However, it is true for metric spaces. (In fact, it holds for any first-countable space.)