Topology/Metric Spaces

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Before we beginEdit

Before we discuss topological spaces in their full generality, we will first turn our attention to a special type of topological space, a metric space. This abstraction has a huge and useful family of special cases, and it therefore deserves special attention. Also, the abstraction is picturesque and accessible; it will subsequently lead us to the full abstraction of a topological space.

Metric SpaceEdit

DefinitionEdit

A metric space is a Cartesian pair   where   is a non-empty set and  , is a function which is called the metric which satisfies the requirement that for all  :

  1.   if and only if  
  2.   (symmetry)
  3.   (triangle inequality)

Note that some authors do not require metric spaces to be non-empty. We annotate   when we talk of a metric space   with the metric  .

ExamplesEdit

  • An important example is the discrete metric. It may be defined on any non-empty set X as follows  
  • On the set of real numbers  , define   (The absolute distance between x and y).
    To prove that this is indeed a metric space, we must show that d is really a metric. To begin with,   for any

real numbers   and  .

    •  .
    •  
    •  
  • On the plane   as the space, and let  

This is the euclidean distance between   and  ).

  • We can generalize the two preceding examples. Let   be a normed vector space (over   or  ). We can define the metric to be:  . Thus every normed vector space is a metric space.
  • For the vector space   we have an interesting norm. Let   and   two vectors of  . We define the p-norm:  . For each p-norm there is a metric based on it. Interesting cases of p are:
    •  . The metric is  
    •  . The metric is good-old Euclid metric  
    •  . This is a bit surprising:  

      As an exercise, you can prove that   thus justifying the definition of  .

  • The Hilbert space is a metric space on the space of infinite sequences {ak} such that
     
    converges, with a metric d({ai}, {bi})= .
  • The concept of the Erdős number suggests a metric on the set of all mathematicians. Take x and y to be two mathematicians, and define d(x, y) as 0 if x and y are the same person; 1 if x and y have co-authored a paper; n if the shortest sequence  , where each step pairs two people who have co-authored a paper, is of length n; or ∞ if x ≠ y and no such sequence exists.

    This metric is easily generalized to any reflexive relation (or undirected graph, which is the same thing).

    Note that if we instead defined d(x, y) as the sum of the Erdős numbers of x and y, then d would not be a metric, as it would not satisfy  . For example, if x = y = Stanisław Ulam, then d(x, y) = 2.

NoteEdit

Throughout this chapter we will be referring to metric spaces. Every metric space comes with a metric function. Because of this, the metric function might not be mentioned explicitly. There are several reasons:

  • We don't want to make the text too blurry.
  • We don't have anything special to say about it.
  • The space has a "natural" metric. E.g. the "natural" metric for   is the euclidean metric  .

As this is a wiki, if for some reason you think the metric is worth mentioning, you can alter the text if it seems unclear (if you are sure you know what you are doing) or report it in the talk page.

Open BallEdit

MotivationEdit

The open ball is the building block of metric space topology. We shall define intuitive topological definitions through it (that will later be converted to the real topological definition), and convert (again, intuitively) calculus definitions of properties (like convergence and continuity) to their topological definition. We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". Then we can instantly transform the definitions to topological definitions.

DefinitionEdit

Given a metric space   an open ball with radius   around   is defined as the set

 .

Intuitively it is all the points in the space, that are less than   distance from a certain point  .

ExamplesEdit

Why is this called a ball? Let's look at the case of  :  .

Therefore   is exactly   - The ball with   at center, of radius  . In   the ball is called open, because it does not contain the sphere ( ).

The Unit ball is a ball of radius 1. Lets view some examples of the   unit ball of   with different p-norm induced metrics. The unit ball of   with the norm   is:   =   =  

  • The metric induced by   in that case, the unit ball is:  

 

  • The metric induced by   in that case, the unit ball is:  

 

  • The metric induced by   in that case, the unit ball is:  

 

As we have just seen, the unit ball does not have to look like a real ball. In fact sometimes the unit ball can be one dot:

  • The discrete metric, The unit ball is  

Interior of a SetEdit

DefinitionsEdit

Definition: We say that x is an interior point of A iff there is an   such that:  . This intuitively means, that x is really 'inside' A - because it is contained in a ball inside A - it is not near the boundary of A.

Illustration:

Interior Point Not Interior Points
   

Definition: The interior of a set A is the set of all the interior points of A. The interior of a set A is marked  . Useful notations:   and  .

PropertiesEdit

Some basic properties of int (For any sets A,B):

  •  
  •  
  •  
  •  

Proof of the first:
We need to show that:  . But that's easy! by definition, we have that   and therefore  

Proof of the second:
In order to show that  , we need to show that   and  .
The "  " direction is already proved: if for any set A,  , then by taking   as the set in question, we get  .
The "  " direction:
let  . We need to show that  .
If   then there is a ball  . Now, every point y, in the ball   an internal point to A (inside  ), because there is a ball around it, inside A:  .
We have that   (because every point in it is inside  ) and by definition  .
Hint: To understand better, draw to yourself  .

Proof of the rest is left to the reader.

ReminderEdit

  • [a, b] : all the points x, such that  
  • (a, b) : all the points x, such that  

ExampleEdit

For the metric space   (the line), we have:

  •  
  •  
  •  
  •  

Let's prove the first example ( ). Let   (that is:  ) we'll show that   is an internal point.
Let  . Note that   and  . Therefore  .
We have shown now that every point x in   is an internal point. Now what about the points   ? let's show that they are not internal points. If   was an internal point of  , there would be a ball  . But that would mean, that the point   is inside  . but because   that is a contradiction. We show similarly that b is not an internal point.
To conclude, the set   contains all the internal points of  . And we can mark  

An Open SetEdit

DefinitionEdit

A set is said to be open in a metric space if it equals its interior ( ). When we encounter topological spaces, we will generalize this definition of open. However, this definition of open in metric spaces is the same as that as if we regard our metric space as a topological space.

Properties:

  1. The empty-set is an open set (by definition:  ).
  2. An open ball is an open set.
  3. For any set B, int(B) is an open set. This is easy to see because: int(int(B))=int(B).
  4. If A,B are open, then   is open. Hence finite intersections of open sets are open.
  5. If   (for any set if indexes I) are open, then their union   is open.

Proof of 2:
Let   be an open ball. Let  . Then  .
In the following drawing, the green line is   and the brown line is  . We have found a ball to contain   inside  .  

Proof of 4:
A, B are open. we need to prove that  . Because of the first propriety of int, we only need to show that  , which means  . Let  . We know also, that   from the premises A, B are open and   . That means that there are balls:  . Let  , we have that  . By the definition of an internal point we have that   (  is the required ball).

Interestingly, this property does not hold necessarily for an infinite intersection of open sets. To see an example on the real line, let  . We then see that   which is closed.

Proof of 5:
Proving that the union of open sets is open, is rather trivial: let   (for any set if indexes I) be a set of open sets. we need to prove that  : If   then it has a ball  . The same ball that made a point an internal point in   will make it internal in  .

Proposition: A set is open, if and only if it is a union of open-balls.
Proof: Let A be an open set. by definition, if   there there a ball  . We can then compose A:  . The equality is true because:   because  .   in each ball we have the element   and we unite balls of all the elements of  .
On the other hand, a union of open balls is an open set, because every union of open sets is open.

ExamplesEdit

  • As we have seen, every open ball is an open set.
  • For every space   with the discrete metric, every set is open.

Proof: Let   be a set. we need to show, that if   then   is an internal point. Lets use the ball around   with radius  . We have  . Therefore   is an internal point.

  • The space   with the regular metric. Every open segment   is an open set. The proof of that is similar to the proof that  , that we have already seen.

TheoremEdit

In any metric space X, the following three statements hold:

1) The union of any number of open sets is open.
Proof: Let   be a collection of open sets, and let
 . Then there exists a   such that  .
So there exists an   such that  . Therefore
 .
2) The intersection of a finite number of open sets is open.
Proof: Let  , where   is a finite collection of open sets.
So   for each  . Let  . For each  , there exists an   such that  . Let  { }. Therefore   and  .
3) The empty set and X are both open.

TheoremEdit

In any metric space X, the following statements hold:

1) The intersection of any number of closed sets is closed.
2) The union of a finite number of closed sets is closed.

ConvergenceEdit

DefinitionEdit

First, Lets translate the calculus definition of convergence, to the "language" of metric spaces: We say that a sequence   converges to   if for every   exists   that for each   the following holds:  .
Equivalently, we can define converges using Open-balls: A sequence   converges to   If for every   exists   that for each   the following holds:  .

The latter definition uses the "language" of open-balls, But we can do better - We can remove the   from the definition of convergence, thus making the definition more topological. Let's define that   converges to   (and mark  ) , if for every ball   around   , exists   that for each   the following holds:  .   is called the limit of the sequence.

The definitions are all the same, but the latter uses topological terms, and can be easily converted to a topological definition later.

PropertiesEdit

  • If a sequence has a limit, it has only one limit.
    Proof Let a sequence   have two limits,   and  . If they are not the same, we must have  . Let   be smaller than this distance. Now for some  , for all  , it must be the case that both   and   by virtue of the fact   and   are limits. But this is impossible; the two balls are separate. Therefore the limits are coincident, that is, the sequence has only one limit.
  • If  , then almost by definition we get that  . (  Is the sequence of distances).

ExamplesEdit

  • In   with the natural metric, The series   converges to  . And we note it as follows:  
  • Any space, with the discrete metric. A series   converges, only if it is eventually constant. In other words:   If and only if, We can find   that for each  ,  
  • An example you might already know:

The space   For any p-norm induced metric, when  . Let  . and let  .
Then,   If and only if  .

Uniform ConvergenceEdit

A sequence of functions   is said to be uniformly convergent on a set   if for any  , there exists an   such that when   and   are both greater than  , then   for any  .

Closed SetsEdit

ClosureEdit

Definition: The point   is called point of closure of a set   if there exists a sequence  , such that  

An equivalent definition using balls: The point   is called point of closure of a set   if for every open ball  , we have  .
The proof is left as an exercise.

Intuitively, a point of closure is arbitrarily "close" to the set  . It is so close, that we can find a sequence in the set that converges to any point of closure of the set.

Example: Let A be the segment  , The point   is not in  , but it is a point of closure: Let  .   ( , and therefore  ) and   (that's because  ).

Definition: The closure of a set    , is the set of all points of closure. The closure of a set A is marked   or  .

Note that  . a quick proof: For every  , Let  .

ExamplesEdit

For the metric space   (the line), and let   we have:

  •  
  •  
  •  
  •  

Closed setEdit

Definition: A set   is closed in   if  .
Meaning: A set is closed, if it contains all its point of closure.


An equivalent definition is: A set   is closed in   If for every point  , and for every Ball  , then  .
The proof of this definition comes directly from the former definition and the definition of convergence.

PropertiesEdit

Some basic properties of Cl (For any sets  ):

  •  
  •  
  •  
  •   is closed iff  
  • While the above implies that the union of finitely many closed sets is also a closed set, the same does not necessarily hold true for the union of infinitely many closed sets. To see an example on the real line, let  . We see that   fails to contain its points of closure,  

This union can therefore not be a closed subset of the real numbers.

The proofs are left to the reader as exercises. Hint for number 5: recall that  .

Open vs ClosedEdit

That is, an open set approaches its boundary but does not include it; whereas a closed set includes every point it approaches. These two properties may seem mutually exclusive, but they are not:

  • In any metric space  , the set   is both open and closed.
  • In any space with a discrete metric, every set is both open and closed.
  • In  , under the regular metric, the only sets that are both open and closed are   and  . However, some sets are neither open nor closed. For example, a half-open range like   is neither open nor closed. As another example, the set of rationals is not open because an open ball around a rational number contains irrationals; and it is not closed because there are sequences of rational numbers that converge to irrational numbers (such as the various infinite series that converge to  ).

Complementary setEdit

A Reminder/Definition: Let   be a set in the space  . We define the complement of  ,   to be  .

A Quick example: let  . Then  .

The plot continues...Edit

A very important Proposition: Let   be a set in the space  . Then, A is open iff   is closed.
Proof: ( ) For the first part, we assume that A is an open set. We shall show that  . It is enough to show that   because of the properties of closure. Let   (we will show that  ).
for every ball   we have, by definition that (*) . If the point is not in   then  .   is open and therefore, there is a ball  , such that:  , that means that  , contradicting (*).
( ) On the other hand, Lets a assume that   is closed, and show that   is open. Let   be a point in   (we will show that  ). If   is not in   then for every ball   we have that  . That means that  . And by definition of closure point   is a closure point of   so we can say that  .   is closed, and therefore   That contradicts the assumption that  

Note that, as mentioned earlier, a set can still be both open and closed!

On  Edit

The following is an important theorem characterizing open and closed sets on  .
Theorem: An open set O in R1 is a countable union of disjoint open intervals.
Proof: Let  . Let   and let  . There exists an open ball (x-ε,x+ε) such that (x-ε,x+ε) ⊆ O because O is open. Thus, a≤x-ε and b≥x+ε. Thus, x ∈(a,b). The set O contains all elements of (a,b) since if a number is greater than a, and less than x but is not within O, then a would not be the supremum of {t|t∉O, t<x}. Similarly, if there is a number is less than b and greater than x, but is not within O, then b would not be the infimum of {t|t∉O, t>x}. Thus, O also contains (a,x) and (x,b) and so O contains (a,b). If y≠x and y∈(a,b), then the interval constructed from this element as above would be the same. If y<a, then inf{t|t∉O, t>y} would also be less than a because there is a number between y and a which is not within O. Similarly if y>b, then sup{t|t∉O, t<y} would also be greater than b because there is a number between y and b which is not within O. Thus, all possible open intervals constructed from the above process are disjoint. The union of all such open intervals constructed from an element x is thus O, and so O is a union of disjoint open intervals. Because the rational numbers is dense in R, there is a rational number within each open interval, and since the rational numbers is countable, the open intervals themselves are also countable.

Examples of closed setsEdit

  1. In any metric space, a singleton   is closed. To see why, consider the open set,  . Let  . Then  , so  . Let  . Then  . So   is open, and hence   is closed.
  2. In any metric space, every finite set   is closed. To see why, observe that   is open, so   is closed.
  3. Closed intervals [a,b] are closed.
  4. Cantor Set Consider the interval [0,1] and call it C0. Let A1 be equal {0,  } and let dn =  . Let An+1 be equal to the set An∪{x|x=a+2dn, a∈An}. Let Cn be  {[a,a+dn]}, which is the finite union of closed sets, and is thus closed. Then the intersection   is called the Cantor set and is closed.

ExercisesEdit

  1. Prove that a point x has a sequence of points within X converging to x if and only if all balls containing x contain at least one element within X.
  2. In   the only sets that are both open and closed are the empty set, and the entire set. This is not the case when you look at  . Give an example of a set which is both open and closed in  .
  3. Let   be a set in the space  . Prove the following:
    1.  
    2.  

ContinuityEdit

DefinitionEdit

Let's recall the idea of continuity of functions. Continuity means, intuitively, that you can draw a function on a paper, without lifting your pen from it. Continuity is important in topology. But let's start in the beginning:

The classic delta-epsilon definition: Let   be spaces. A function   is continuous at a point   if for all   there exists a   such that: for all   such that  , we have that  .

Let's rephrase the definition to use balls: A function   is continuous at a point   if for all   there exists   such that the following holds: for every   such that   we have that  . Or more simply:  

Looks better already! But we can do more.

Definitions:

  • A function is continuous in a set S if it is continuous at every point in S.
  • A function is continuous if it is continuous in its entire domain.

Proposition: A function   is continuous, by the definition above   for every open set   in  , The inverse image of  ,  , is open in  .
Note that   does not have to be surjective or bijective for   to be well defined. The notation   simply means  .

Proof: First, let's assume that a function   is continuous by definition (The   direction). We need to show that for every open set  ,   is open.

Let   be an open set. Let  .   is in   and because   is open, we can find and  , such that  . Because f is continuous, for that  , we can find a   such that  . that means that  , and therefore,   is an internal point. This is true for every   - meaning that all the points in   are internal points, and by definition,   is open.

( )On the other hand, let's assume that for a function   for every open set  ,   is open in  . We need to show that   is continuous.

For every   and for every  , The set   is open in  . Therefore the set   is open in  . Note that  . Because   is open, that means that we can find a   such that  , and we have that  .

The last proof gave us an additional definition we will use for continuity for the rest of this book. The beauty of this new definition is that it only uses open-sets, and there for can be applied to spaces without a metric, so we now have two equivalent definitions which we can use for continuity.

ExamplesEdit

  • Let   be any function from any space  , to any space  , were   is the discrete metric. Then   is continuous. Why? For every open set  , the set   is open, because every set is open in a space with the discrete metric.
  • Let   The identity function.   is continuous: The source of every open set is itself, and therefore open.

ExerciseEdit

  1. Prove that a function   is continuous   for every closed set   in  , The inverse image of  ,  , is closed in  .

Uniform ContinuityEdit

In a metric space X, function from X to a metric space Y is uniformly continuous if for all  , there exists a   such that for all  ,   implies that  .

IsometryEdit

An isometry is a surjective mapping  , where   and   are metric spaces and for all  ,  .

In this case,   and   are said to be isometric.

Note that the injectivity of   follows from the property of preserving distance:

 
 
 
 

So an isometry is necessarily bijective.

ExercisesEdit

  1. Show that a set is a metric open set iff it is a (possibly infinite) union of open balls.
  2. Show that the discrete metric is in fact a metric.


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