Perhaps the best way to learn basic ideas about topology is through the study of point set topology. The concepts and definitions can be illuminated by means of examples over a discrete and small set of elements. The early champions of point set topology were Kuratowski in Poland and Moore at UT-Austin. For a long time Point Set topologists were isolated from the Combinatorial Topologists.

In fact the word topology (although used in the German title of an article by Listing in 1847), evidently did not become popular until Lefshetz introduced the English word in a 1930 monograph. In the mid-late 20's it was recognized that analysis situs (topology) had many equivalent expositions, and the Point Set topologists began talking to the Combinatorial Topologists.

One of the best books for rapid assimilation of point set topology is the Schaum's Outline Series, "General Topology" by S Lipschutz. I have an old copy (1965). I advise you to review Chapter 1, then skip to chapters 5, 6, and 7.

In that which follows, four different topologies will be defined over a set of five elements, {a,b,c,d,e}. Then the topological definitions of

• Open sets
• Closed Sets
• Limit Points
• Closure
• Boundary
• Interior
• Exterior
• Continuity

will be defined and exemplified for each of the four point set topologies. It should become apparent that the same sets of points can have different topologies imposed as a set of constraints on the same elements. It is the topology that allows the concepts of boundary, closure and limit points to be define. In that which follows, the applications of these ideas will done for systems of differential forms, rather than systems of points.

The symbol ^ will be used to represent the set intersection and the symbol U
will be used to represent set union. (using the HTML language of the Internet)

Closed and Open Sets

Consider a set of elements {a,b,c,d,e} and a combinatorial process which is symbolized, for example, by the brackets (ab) or (ade). Construct all possible combinations, and include the null set, 0. Define X = (abcde).

Now from the set of all possible combinations, it is possible to select many subset collections. Certain of these subset collections have the remarkable property of logical closure. As an example, consider the subset collection, or class of subsets, given by:

Note that the intersection of (a) with (ab) is a, (a^(ab) = a), which is an element of the collection, and the intersection of (ab) with (bcd) is b which is also an element of the collection. In fact, the intersection of every element of T1(closed) with every other element of T1(closed) produces one of the original seven elements of the collection, T1closed. In other words, the process of set intersection acting on any number of elements is closed with respect to T1(closed).

Now also note that the union of any two elements of the set is also contained within the set. The idea that a closed algebra can be built upon the notions of union and intersection, and that this algebra be a division algebra, is at the heart of the theory of logic. This idea of logical closure with respect to arbitrary intersection and finite union is said to define a topology, T1(closed), of closed sets.

Definition: A topology T1(closed) on a set X is a collection or class of subsets that obeys the following axioms:

• A1(closed): X and the null set 0 are elements of the collection.
• A2(closed): The arbitrary intersection of any number of elements of the collection belongs to the collection.
  
• A3(closed): The arbitrary union of any pair of elements of thecollection belongs to the collection.
  

The elements of the collection, T1(closed), are defined to be "closed" sets. The compliments of the closed sets are defined as "open" sets. The open sets of the topology are the collection of subsets given by

It is important to note that the same set of all combinations of subsets can support many topologies. For example, the subsets of the collection,

are closed with respect to both logical intersection and union. Hence T2(closed) is a different topology built on the same set of points, X. The open sets of this topology are

The compliments of closed sets are defined to be "open" sets and they too can be used to define a topology. A subset can be both open, or closed, or both, or neither, relative to a specified topology. For example, with respect to the topology given by the closed sets,

(bcde) is both open and closed, and the set (bc) is neither open nor closed.

The topology of closed sets given by the collection,

has its dual as the topology of open sets

Note that this topology, T4, is a refinement of the topology, T3, in that it contains additonal closed (or open) sets.

Remarks: In the definition of a topology when the number of elements of the set in not finite, the logical intersection of open sets is restricted to any pair, and the logical union of closed sets to restricted to any pair. There are many other ways to define a topology, but the concepts always come back to the idea of logical closure.

Limit Points

The next idea to be presented is the concept of a limit point. A standard definition states that a point p is a limit point of a subset, A, iff every open set that contains p contains another point of A. Given a subset, A, each point of X must be tested to see if it is a limit point of A relative to the topology specified on the points. If A is a singleton, it can have no limit points, for there are no other points of A. It follows that the limit points of a limit point (a singleton) is the null set. If the limit point of A consists of the singletons or points symbolized by dA, then d(dA) = 0. The set of limit points as a collection of singletons, {a,b,c..} will be denoted by dA, where the union of all limit points will be denoted by A`. The symbol d may be viewed as a limit point operator; the symbol d. when applied to a set, A, means that each point p of the domain is tested against the specified topology to see if another point of A is included in each open set of the topology.

Consider the subset A = (ab) and the topology given by T1(open). Now test each point relative to the collection T1(open):

• The point a is not a limit point of (ab) because the open set (acde) does not contain b;
  dA = 0 at a.
  
• The point b is not a limit point of (ab) because the open set (bcde) does not contain a;
  dA = 0 at b.
  
• The point c is not a limit point of (ab) because the open set (cde) does not contain either a or b;
  dA = 0 at c
  
• The point d is not a limit point of (ab) because the open set (cde) does not contain either a or b;
  dA = 0 at d.
  
• The point e is not a limit point of (ab) because the open set (cde) does not contain either a or b;
  dA = 0 at e.
  

Now make the same tests with regard to the same subset A = (ab), but this time relative to the topology given by T2(open).

• The point a is not a limit point of (ab) because the open set (a) is a singleton;
  dA = 0 at a.
  
• The point b is a limit point of (ab) because both the open sets that contain b also contain a;
  dA # 0 at b.
  
• The point c is a limit point of (ab) because the open set X, the only open set that contains c, contains a and b which are points of A;
  dA # 0 at c.
  
• The point d is a limit point of (ab) because the open set X , the only open set that contains d, contains another point of A;
  dA # 0 at d.
  
• The point e is a limit point of (ab) because the open set X, the only open set that contains e, another point of A; dA # 0 at e.
  

Hence, the points b,c,d and e are limit points of A = (ab) relative to the topology T2(open).

With respect to the topology of T4(open),

test for limit points of the set (ab):

• The point a is not a limit point of (ab) because the open set a is a singleton;
  dA = 0 at a.
  
• The point b is not a limit point of (ab) because the open set (bcde) does not contain a;
  dA = 0 at b.
  
• The point c is not a limit point of (ab) because the open set (cd) which contains c does not contain a or b; dA = 0 at c.
• The point d is not a limit point of (ab) because the open set (cd) which contains d does not contain a or b; dA = 0 at d.
• The point e is a limit point of (ab) because the open set (bcde) which contains e contains another point b of A; dA # 0 at e.

The limit set of (ab) relative to T4(open) becomes

Note that the set of limit points as a collection, or a class of sets, may or may not have limit points. If the limit set is a singleton, then the limit points of the set of limit points is the null set. However, consider the limit set A` = {bcde} of the set (ab) relative to the topology T2. Then the limit points of A` are the points (b,c,d,e). In other words dA` 0 necessarily, but ddA` = 0.

Closure

The closure of a set is defined to be the union of the set and its limit points. Note that a closed set contains its limit points, if any exist. In the examples given above the closure of the set A = (ab) relative to the topology T1 is equal to the union of A = (ab) and its limit points, which is the null set.

A~ = A U A` = (ab).

Note that A = (ab) is a closed set, and has no limit points relative to the topology T1(open). However, the closure of A relative to the topology T2(open) is

A~ = A U A` = (ab) U (bcde) = (abcde) .

which is the whole set. When the closure of a subset is the whole set X, the subset is said to be dense in X relative to the specified topology.

The closure of (ab) relative to the topology T4(open) is

A~ = A U A` = (ab) U (e) = (abe).

Note that the closure of a subset is equal to the smallest closed set that contains the subset. Every closed set is its own closure. A closed set may or may not have limit points, but if it does have limit points they are contained within the (closed) set.

Continuity

Now comes a major issue of this lecture. Continuity of a transformation is defined relative to the topologies that may exist on the initial and final states. Let the set of points X with topology T1(open) be mapped into the set of points Y with the topology T2(opn) Then the map is continuous iff the closure of every subset of the initial state relative to T1 is included in the closure of the image of the final state relative to the topology T2.

Another test for continuity is given by the statement that the inverse image of every open set of Y relative to T2 is an open set of X relative to the topology T1.

Later on, the first definition will be used to prove that any topology built on subsets of exterior forms with C2 (twice differentiable) coefficients will be continuously transformed by evolutionary processes that are generated by the Lie convective derivative with respect to C2 vector fields. For the present, the second definition will be used in terms of simple point set topological systems.

As a first example consider the transformations given on X to Y by the following diagram:

Continuous example

Discontinuous example

Homeomorphic example

(equivalent topologies)

Interior

When emphasis is placed on open sets rather that closed sets, other ideas come to the forefront. In particular, the concept dual to the notion of closure is the concept of interior. While closure asks for the smallest closed set that covers any specified subset, the idea of interior asks for the largest open set included in the specified subset. The interior of a set can be empty (for there may be no open sets other than the null set contained within the specified set)!

For example, the subset (ab) has no interior relative to the topology T1(open). However, the interior of (ab) is itself, (ab), relative to the topology T2(open), because (ab) is an open set in this topology! Relative to the topology T4(open), the interior of (ab) is the singleton, (a):

• Int A = {0} relative to T1'
• Int A = (ab) relative to T2
• Int A = (a) relative to T4.

The set (abe) has an interior (ae) relative to T1(open) and an interior (ab) relative to the topology T2(open).

The exterior of a specified set is the interior of the compliment of the specified set. The compliment of (ab) is the set (cde) which has the interior (cde) relative to T1(open) and has no interior relative to the topology T2(open). Relative to the topology T4(open), the exterior of (ab) is the set (cd).

• Ext A = {cde} relative to T1
• Ext A = (0) relative to T2
• Ext A = (cd) relative to T4.

The Boundary

The points that make up the boundary of a subset are union of those points that are not included in the interior or the exterior. However, the union of the points that make up the boundary may have subsets that are not connected. Consider a solid disk. The points that make up the rim of the disk forms its boundary.

Now punch a hole in the disk. The collection of points that make up the outer rim and the inner hole now form the boundary of the disk. The two sets of boundary points are not connected.

Similar to the limit point operator, d, a boundary operator, (some books use ) , may be defined in terms of a procedure, such that when is applied to the set A, it implies that a test is made at each point p to see if p is an element of the interior or of the exterior of the selected subset A. If the test fails then A # 0 and the point is a boundary point. It the point p is an element of the exterior or interior of A, then A = 0 at the point p. The bA, of the set A is defined as the union of all boundary points.

As a first example, consider again the set A = (ab) and the T1 topology. The set A=(ab) has no interior, but the exterior of (ab) is the set (cde), and therefore the boundary set, bA, consists of the union of 

the points (a,b). In this first example, then

A` = {0}, but bA = (ab) included in A~.

It follows that

A ^ A` = 0,

A ^ bA # 0.

The boundary set exists even though the limit set does not!

Relative to T2(open), the set (ab) has an interior set (ab), no exterior set, but a boundary set is bA = (cde). In this case the boundary is included in the closure,

A` = (bcde), bA = (cde) included in A~ = (abcde),

and

A ^ A` 0,

A ^ bA = 0.

Although all boundary points are limit points, there exist limit points that are not elements of the boundary.

Relative to T4(open), the set (ab) has an interior set (a), and exterior set (cd) and a boundary set (be),

A` = (e), bA = (be),

A ^ A` = 0,

A ^ bA # 0.

It is apparent that the boundary points contain limit points, but there are boundary points which are not limit points!.

In all cases, note that the union of the interior and the boundary is equal to the union of the set and its limit points. The boundary is always included in the closure, but the boundary may contain points which are not limit points.

A ~ = IntA U bA = A U A`.

These examples point out that there exist certain correspondences between limit points and boundaries, but they are not necessarily the same concept. Much of current physical theory has emphasized the boundary and open set point of view, while in this lecture the emphasis is on the limit points and closure point of view. It will become evident that these concepts are at the heart of the differences between contravariant and covariant concepts in physical theories, an idea that ultimately expresses itself in the differences between the particle or wave perspective of physics. In topology, these notions are at the heart of the differences between Homology and Cohomology (which will be discussed in detail later). In this text, the Cohomological point of view is emphasized.

If a set has the property that its intersection with its limit set is empty, then the set is said to be isolated. This idea of isolation, whereby A A` = 0, will be translated into the Cartan statement, A^dA = 0, in the next chapter. The physical significance ot the topological concept of isolation will be correlated with the Caratheodory statement of the existence of inaccessible states in a thermodynamic system, and to the notion of Frobenius complete integrability for a laminar, non-chaotic flow. The concept of isolation is a topological property. and its compliment is a necessary condition for chaos. The observation of a flow transforming from a laminar state (isolated) to a turbulent state (non-isolated) is an observation of topological evolution.

It should be mentioned that with respect to diffeomorphic transformations, or more simply those transformations that preserve pure geometrical properties, the differences between contravariant and covariant concepts cannot be distinguished. Further note that the existence of a metric implies that the contravariant concepts can be converted into covariant concepts, and their possible differences are masked into an alias-alibi format; that is, there are no measurable differences between the two concepts. However, with respect to an aging process, the behavior of the two concepts is observably different. The differences between the behavior of contravariant and covariant concepts may be interpreted as the existence of topological evolution.