What is a Topology?

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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

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The following reading is suggested to help supplement this lesson.

Definition (topology)

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

  1.   (where   denotes the power set of X)
  2. For   we have  
  3. 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

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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

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See also Wikipedia articles:

Definition (open set, closed set,neighborhood)

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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)

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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)

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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)

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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)

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

Reading supplement

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See also Wikipedia articles:

Lesson Exercises

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  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

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  1. where