Topology/Lesson 1

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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.

Definition (topology)[edit]

Let be a set. Then a topology on is a set (where denotes the power set of X) such that the following two conditions hold.

  1. For we have
  2. 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 and 2 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

  1. for any we have and
  2. 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]

  1. 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?
  2. Can you find a formula for the number of topologies on an -point set?
  3. 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
  4. 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
  5. 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
  6. It is clear from the definition that Show that if then and
  7. Show that and that Use these facts to show that is open if and only if and is closed if and only if
  8. Is it true that for any set that Give a proof or a counterexample.
  9. Show that the collection [1] of open intervals is a basis for a topology on This is called the standard topology on

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Notes[edit]

  1. where