# Topology/Lesson 1

## What is a Topology?[edit]

The word "topology" has two meanings: it is both the name of a mathematical subject and the name of a mathematical structure. A *topology on a set * (as a mathematical strucure) is a collection of what are called "open subsets" of satisfying certain relations about their intersections, unions and complements. In the basic sense, Topology (the subject) is the study of structures arising from or related to topologies.

### Reading Assignment[edit]

- The following reading is suggested to help supplement this lesson.

- Wikibooks Topology book:
- Wikipedia articles:

### Definition (topology)[edit]

Let be a set. Then a *topology* on is a set such that the following conditions hold.

- (where denotes the power set of X)
- For we have
- For finite sets we have

The set together with the topology is called a *topological space* (or simply a *space*) and is commonly written as the pair Or, when is understood it may be omitted and we will simply say that is a topological space.

### Examples[edit]

Here are some very simple examples of topological spaces. For these examples, can be any set.

- Discrete topology
- The collection is called the
*discrete topology*on - Indiscrete topology
- The collection is called the
*indiscrete topology*or*trivial topology*on - Particular point topology
- Given a point the collection is called the
*particular-point topology*on

It is left as an exercise to verify that each of these three collections does indeed satisfy the axioms of a topology (conditions 1,2,3 in the definition above).

#### Reading supplement[edit]

See also Wikipedia articles:

### Definition (open set, closed set,neighborhood)[edit]

Suppose that is a topological space.

- Open set
- A set is
*open*if - Closed set
- A set is
*closed*if - Neighborhood
- For a point a set is a
*neighborhood of*if there is an open set such that

### Definition (closed topology)[edit]

- Alternate definition of a topology

Suppose that Then is a *closed topology* if

- for any we have and
- for any finite collection we have

Show that for any set the collection is a topology on if and only if the collection is a closed topology on

### Definition (interior, closure)[edit]

Let be a space and let

- Interior
- The
*interior of*(denoted ) is defined to be the union of all open sets contained in In other words, - Closure
- The
*closure of*(denoted ) is defined to be the intersection of all closed sets containing That is,

### Definition (basis)[edit]

Let be a space. Then a collection is a *basis* if for any point and any neighborhood of there is a basis element such that

The benefit of talking about a basis is that sometimes describing every open set is unwieldy. For example, describing an open set in the Euclidean plane would be difficult, but describing a basis is very easy. A basis of open sets in the plane is given by "open rectangles". That is forms a basis.

Once a basis is determined, a set is open if it is the union of basis elements. That is, if is a basis, then the topology is given by

### Definition (compact)[edit]

Let be a topological space. Then a set is *compact* if and only if every open cover of has a finite subcover.

#### Reading supplement[edit]

See also Wikipedia articles:

## Lesson Exercises[edit]

- Let be a three-point set. Then there are different subsets of How many of these are topologies on In other words, how many different 3-point topologies are there?
- Can you find a formula for the number of topologies on an -point set?
- Suppose that is such that for any there is a set containing and that for any two sets such that there is a set such that Show that the collection is a topology on and that is a basis for
- Let be such that for all there is a set which contains Then show that the collection is a basis for a topology on (using the criterion given in exercise 3). In this case, we call a
*subbasis*for - A basis for a topology is said to be
*minimal*if any proper collection is not a basis for Given a set find a minimal basis for the discrete topology - It is clear from the definition that Show that if then and
- Show that and that Use these facts to show that is open if and only if and is closed if and only if
- Is it true that for any set that Give a proof or a counterexample.
- Show that the collection
^{[1]}of open intervals is a basis for a topology on This is called the*standard topology on*

## Notes[edit]

- ↑ where