# Topology/Topological Spaces

In this section, we will define what a topology is and give some examples and basic constructions.

## Motivation[edit]

In Abstract Algebra, a *field* generalizes the concept of operations on the real number line. This general definition allows concepts about quite different mathematical objects to be grasped intuitively by comparison with the real numbers. Likewise, the concept of a **topological space** is concerned with generalizing the structure of sets in Euclidean spaces. Of course, for many topological spaces the similarities are remote, but aid in judgment and guide proofs. Interesting differences in the structure of sets in Euclidean space, which have analogies in topological spaces, are connectedness, compactness, dimensionality, and the presence of "holes".

If we begin with an arbitrary set, it may not be immediately obvious what is needed to imbue it with an interesting structure. One possibility might be to define a metric on the set, but as it turns out, requiring a metric is overly restrictive. In fact, there are many equivalent ways to define what we will call a **topological space** just by defining families of subsets of a given set. The properties of the topological space depend on the number of subsets and the ways in which these sets overlap. Topological spaces can be fine or coarse, connected or disconnected, have few or many dimensions.

The most popular way to define a topological space is in terms of open sets, analogous to those of Euclidean Space. (In Euclidean space, an open set is intuitively seen as a set that does not contain its "boundary").

## Definition of a topological space[edit]

Given a set , a **topology** on is a collection of subsets of with the following properties:

- The empty set and are both in .
- The union of any collection of open sets is an open set. That is, .
- The intersection of any
*finite*collection of open sets is an open set. That is, .

The pair is called a **topological space**. If the topology is clear or does not need an explicit name (since we can just refer to sets in the topology as open sets), then we just say that is a topological space.

## Examples of topological spaces[edit]

For any set , there are two topologies we can always define on :

- The
- the topology consisting of all subsets of a set .**Discrete topology** - The
(also known as the**Indiscrete topology**) - the topology consisting of just and the empty set, .**trivial topology**

### Metric Topology[edit]

Given a metric space , its **metric topology** is the topology induced by using the set of all open balls as the base. One can also define the topology induced by the metric, as __the set of all open subsets defined by the metric__. We denote the topology induced from the metric d with

This forms a topological space from a metric space.

If for a topological space , we can find a metric , such that , then the topological space is called metrizable.

### The usual topology on the real numbers[edit]

We can define a topology on by defining to be in if for every point , there is an such that . We call this topology the **standard topology**, or **usual topology** on .

### The cofinite topology on any set[edit]

Let be a non-empty set. Define to be the collection of all subsets of satisfying the following:

- Either
- Or is finite.

Then is a topology on called **the cofinite topology** (or "finite complement topology") on . Further, this topology turns out to be discrete if and only if is finite.

### The cocountable topology on any set[edit]

Let be a non-empty set. Define to be the collection of all subsets of satisfying the following:

- Either
- Or is countable.

Then is a topology on called **the cocountable topology** (or "countable complement topology") on . Further, this topology turns out to be discrete if and only if is countable.

## Sets in topological spaces[edit]

Let be a topological space. There are many types of sets we can define on

- The complement of a set A in X, denoted by , is (that is, the entire space except for A).
- A subset is called
**closed**if the set is open. Notice that the intersection of any non-zero number of closed sets is closed and the union of finitely many closed sets is closed. - Note also that a set can be both
*closed*and*open*. The trivial examples are the empty set and the entire set , each of which is both closed and open. By definition, is open, so its complement, , is closed. But , by definition, is an open set, so is both open and closed. - A set is called a
**neighborhood**of a point , if there is an open set such that

We now investigate some commonly occurring sets in the study of Topology.

### Definition[edit]

In a topological space, a * set* is a countable intersection of open sets. A * set* is a countable union of closed sets.

### Theorem[edit]

The complement of a set is , and vice versa.

**Proof:**

Let A be a set and let . Then A is a countable union of closed sets, such that is closed for all n. Then . Since is closed, is open, so we have a countable intersection of open sets. Hence is .

The entirely similar proof of the other implication is left to the reader.

### Theorem[edit]

In any metric space, a closed set is a set.

Proof:

Let X be a metric space and let .

Define . Observe that is open for any n, and hence the union is open. Now our goal is to show that to show that a closed set is the intersection of countably many open sets.

:

Let . Then intersects A at some which implies . This is true for any n so .

:

Let and . Then such that . So in A such that , which implies . Thus .

Therefore and is a set.

### Theorem[edit]

In usual , is a set.

**Proof:**

Since with the usual topology is a metric space, every singleton such that is closed. Thus, we have a countable union of closed sets, and hence is a set.

## Exercises[edit]

- Prove the following are topologies:
- The discrete topology on any set.
- The indiscrete topology on any set.
- The cofinite topology on any set.
- The cocountable topology on any set.

- Show that the cofinite (respectively, cocountable) topology on a set equals the discrete topology if and only if is finite (respectively, countable).
- Prove that a set is open if and only if for every element within the set, there is a neighborhood contained within the set.
- Show that the discrete topology is the topology induced by the discrete metric. (This is also a splendid way of remembering the discrete and the indiscrete topology)