Topological space. Topology. Open and closed sets. Neighborhood. Interior, exterior, limit, boundary, isolated point. Dense, nowhere dense set.

Def. Topology (on a set). A topology on a set X is a collection τ of subsets of X, satisfying the following axioms:

(1) The empty set and X are in τ

(2) The union of any collection of sets in τ is also in τ

(3) The intersection of any finite number of sets in τ is also in τ

The elements of X are usually called points, although they can be any mathematical objects. The sets in τ are called open sets and their complements in X are called closed sets. Subsets of X may be either closed or open, neither closed nor open, or both closed and open. A set that is both closed and open is called a clopen set. The sets X and ∅ are both open and closed.

Def. Topological space. A set X for which a topology τ has been specified is called a topological space.

It can be shown that axioms (1), (2), and (3) are equivalent to the following two axioms:

(a) The union of any collection of sets in τ is also in τ

(b) The intersection of any finite number of sets in τ is also in τ

A topology on a set X then consists of any collection τ of subsets of X that forms a closed system with respect to the operations of union and intersection.

Notation. Topological space (X, τ). A topological space X with topology τ is often referred to as the topological space (X, τ).

The collection τ of open sets defining a topology on X doesn’t represent all possible sets that can be formed on X. Let π be the set of all possible sets that can be formed on X. The union or intersection of any two sets in π is a set in π. The union or intersection of any two sets in τ is a set in τ. τ represents some subset of π that is closed with respect to the operations of union and intersection. Thus we have here a situation that is analogous to a subgroup of a group or a subspace of a vector space where the elements of some subset of a larger set form a closed system (with respect to an operation) and are thus a closed system embedded within a larger closed system.

For insight and clearer understanding of the concepts look to the point sets of one, two and three dimensional spaces for a model from which to think. Let X be the points of the open interval (a, b) on the real line and τ be the set of all open sets in X. In Fig. 1 is depicted a typical open set, closed set and general set (neither open nor closed) on the interval. The set π corresponds to all possible unions and intersections of general sets in X. The union or intersection of any two open sets in X is open. Thus the collection of all open sets in X form a closed system with respect to the operations of union and intersection. They constitute a subset τ of the collection of all possible sets π in X. The set X with the topology τ represents a topological space.

If we now let X be the closed interval [a, b], the collection of all closed sets in X form a closed system with respect to the operations of union and intersection. They also constitute a subset of the collection of all possible sets π on [a, b]. Thus the set τ of all closed sets in the interval [a, b] provide a topology for X = [a, b]. The set X = [a, b] with the topology τ represents a topological space.

In Fig. 2 is depicted a typical open set, closed set and general set in the plane where dashed lines indicate missing boundaries for the indicated regions. The complement of any closed set in the plane is an open set.

We will now give a few more examples of topological spaces.

1. Let τ be the collection all open sets on R. (where R is the set of all real numbers i.e. the real line). Then τ is a topology on R. The set τ is called the usual topology on R. R with the topology τ is a topological space.

2. Let τ be the collection all open sets on R2 (where R2 is the Cartesian product R R i.e. the plane). Then τ is a topology on R2. The set τ is called the usual topology on R2. R2 with the topology τ is a topological space.

3. Let X be the discrete set

X = {a, b, c, d, e} .

Then the collection of subsets

T1 = {X, ∅ , {a}, {c, d}, {a, c, d}, {b, c, d, e}}

is a topology on X. X with the topology T1 is a topological space. Note that the union and intersection of any two sets in T1 is a set in T1.

From this example we see that the points of X can be discrete points and need not be points of a continuum as was the case in our previous examples. The points need only meet the requirements of our axiomatically-oriented definition to qualify as a topology. Consider now the collection of subsets

T2 = {X, ∅, {a}, {c, d}, {a, c, d}, {b, c, d}} .

This collection of sets is not a topology since the union

{a, c, d} ∪ {b, c, d} = {a, b, c, d}

is not in T2.

4. Let X be the discrete set

X = {a, b, c} .

Then all of the following collections of sets constitute topologies on X:

T1 = {{X, ∅ , {a, b}, {b}, {b, c}}

T2 = {{X, ∅, {a, b}, {b}, {b, c}, {c}}

T3 = {{X, ∅ , {a}, {a, b}}

These sets can be indicated schematically by the diagrams of Fig. 3. We have listed only three topologies but many more could be listed for X = {a, b, c} .

We thus see that a given set X can have many topologies.

5. Let X be any set of points. Then the collection D of all subsets of X is a topology on X. It is

called the discrete topology on X. X with its discrete topology D is called a discrete topological space or simply a discrete space.

6. Let X be any set of points. Then the collection consisting of X and ∅ is a topology on X. It is called the indiscrete topology or trivial topology. X with the indiscrete topology is called an indiscrete topological space or simply an indiscrete space.

7. Let X be the set of points in the plane shown in Fig. 4. Let τ be the collection all open sets on X. Then τ is a topology on X. X with the topology τ is a topological space.

8. Let X be the set of points in space shown in Fig. 5. Let τ be the collection all open sets on X. Then τ is a topology on X. X with the topology τ is a topological space.

9. Let τ be the collection all open-closed sets (a, b] on R. Then τ is a topology on R.

10. Let τ be the collection all closed sets on R. Then τ is a topology on R.

Theorem. Let T1 and T2 be two different topologies on X. Then the intersection T1 T2 is also a topology on X.

A topological space is a generalization / abstraction of a metric space in which the distance concept has been removed. A topological space, unlike a metric space, does not assume any distance idea. We will now give the definition of a number of topological terms as defined for a metric space and then give the definitions of the same terms for a topological space for comparison.

Metric space definitions

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Def. ε-neighborhood of a point. The ε-neighborhood of a point P is the open set consisting of all points whose distance from P is less than ε. We note that the boundary is not included. Another name for ε-neighborhood is open sphere. Other names are spherical neighborhood and ball. The open sphere at point p is denoted by S(p, ε).

Def. ε-deleted neighborhood of a point. The ε-deleted neighborhood of a point P is the ε- neighborhood of the point P minus the point P itself.

Def. Neighborhood of a point. A neighborhood of a point P is any set that contains an ε-neighborhood of P.

Note. We see, from the definitions, that while an ε-neighborhood of a point is an open set a neighborhood of a point may be open, closed or neither open nor closed..

Def. Interior point of a point set. A point P is called an interior point of a point set S if there exists some ε-neighborhood of P that is wholly contained in S.

Def. Boundary point of a point set. A point P is called a boundary point of a point set S if every ε-neighborhood of P contains points belonging to S and points not belonging to S.

Def. Exterior point of a point set. A point P is an exterior point of a point set S if it has some ε-neighborhood with no points in common with S i.e. a ε-neighborhood that lies wholly in , the complement of S. If a point is neither an interior point nor a boundary point of S it is an exterior point of S.

Def. Limit point. A point P is called a limit point of a point set S if every ε-deleted neighborhood of P contains points of S.

Def. Derived set. The set of all limit points of a set S is called the derived set and is denoted by .

Def. Isolated point. A point of a point set in whose neighborhood there is no other point of the set.

Def. Open set. A point set is said to be open if each of its points is an interior point.

Def. Closed set. A point set is called closed if it contains all of it limit points.

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The above definitions were created in reference to a particular model — a model based in two and three dimensional space. The model is the curves, surfaces and solids of two and three dimensional space. These curves, surfaces and solids are conceived as being made up of aggregates of points in a continuum i.e. a curve is viewed as a one-dimensional continuum of points, a surface is viewed as a two-dimensional continuum of points and a solid is viewed as a three-dimensional continuum of points. The above definitions provide tests that let us determine if a particular point in a continuum is an interior point, boundary point, limit point , etc. They define with precision the concepts interior point, boundary point, exterior point , etc in connection with the curves, surfaces and solids of two and three dimensional space. They are terms pertinent to the topology of two or three dimensional space. The terms are intuitive. They make sense. When the idea of a metric space was conceived it was possible to extend these definitions unchanged to a metric space. We note that these definitions were made employing the concept of a neighborhood and the concept of a neighborhood assumes a metric, a distance idea, on the space.

A topological space is an abstract mathematical structure in which a metric is not a qualifying requirement. A wide range of structures can qualify as a topological space ranging from points sets representing continua in two or three dimensional space to discrete sets of isolated points. Yet the terms interior point, boundary point, exterior point, limit point, open set and closed set are defined for this very general structure in its full breadth. How can this be done? It is obvious that for a set of discrete, isolated points these terms can have no meaning remotely close to the usual meaning of the terms. The definitions have been carefully phrased so that they are mostly equivalent to the above metric space definitions for that special case when the topological space corresponds to spaces that do possess a metric (i.e. metric spaces). But what can the meaning of the terms be for a space of discrete points?

The definition of the above terms for a topological space follow. It was necessary to redefine the above concepts for a topological space because a topological space contains no distance notion so the definitions have to be stated in such a way as to avoid a distance idea. When reading the following definitions think in terms of points sets in two or three dimensional space such as those shown in Fig. 4. Don’t think in terms of a discrete point set! If X is the set of points shown in Fig. 4 and τ is the collection of all open sets on X then τ is a topology on X. X with the topology τ is a topological space. Apply the definitions to the interiors, boundaries, etc. of the point set of Fig. 4 for insight and understanding. For that particular case in which a topological space is a metric space the open sets of the topological space consist of the open sets of the metric space.

Topological space definitions

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Def. Open set. Any of the subsets of a topological space X that comprise a topology on X are called open.

Def. Closed set. A subset A of a topological space X is called closed if and only if its complement Ac in X is open i.e. iff X - A is open.

Def. Open neighborhood of a point p ε X. Any open set containing p.

Def. Neighborhood of a point p ε X. Any set that contains an open set that contains p. See Fig. 6.

We thus see from the definition that a neighborhood of a point may be open, closed, neither open nor closed, or both open and closed.

Def. Neighborhood system of a point p ε X. The collection Np of all neighborhoods of p.

Def. Isolated point of a set. Let A be a subset of topological space X. A point in A is called an isolated point of A if it has an open neighborhood which contains no other point of A. See Fig. 7.

Def. Limit point. Let A be a subset of topological space X. A point p in X is called a limit point of A if each of its open neighborhoods contains a point of A different from p i.e. if every open set containing p contains a point of A different from p. See Fig. 8.

Syn. Accumulation point, cluster point, derived point

Def. Derived set. Let A be a subset of topological space X. The derived set of A, denoted by A' or D(A), is the set of all limit points of A.

Def. Closure of a set. Let A be a subset of topological space X. The closure of A, denoted by or , is the intersection of all closed supersets of A (Consider the set of all closed supersets of A. is their intersection.) A point in the closure of A is called a closure point of A.

● The closure of A is the union of the interior and boundary of A, i.e. b(A).

Def. Interior of a set. Interior point. Let A be a subset of topological space X. The interior of A, denoted by A0 or Int A, is the union of all open subsets of A. A point in the interior of A is called an interior point of A. See Fig. 8.

● The interior of A is the largest open subset of A and A is open if and only if A = A0 .

Def. Exterior of a set. Exterior point. Let A be a subset of topological space X. The exterior of A, denoted by Ext A, is the interior of the complement of A i.e. int Ac. A point in the exterior of A is called an exterior point of A.

Def. Boundary of a set. Boundary point. Let A be a subset of topological space X. The boundary of A, denoted by b(A), is the set of points which do not belong to the interior or the exterior of A. A point in the boundary of A is called a boundary point of A. See Fig. 8.

● The boundary of A is given by (where is the closure of A and Ac is the complement of A).

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It is of value to compare the above definitions with those for a metric space. Note how the definitions of neighborhood, limit point, interior point, etc. have been radically restated.

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Theorems

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Theorem 1. In a topological space x, a subset of X is open if and only if its complement is closed i.e. a subset of X is open its complement is closed.

Example 1. The collection

T = {X, ∅ , {a}, {c, d}, {a, c, d}, {b, c, d, e}}

defines a topology on X = {a, b, c, d , e}. The closed subsets of X are

∅, X, {b, c, d, e}, {a, b, e}, {b, e} {a}

i.e. the complements of the open subsets of X. Note that there are subsets of X, such as {b, c, d , e} that are both open and closed and there are subsets of X , such as {a, b}, that are neither open nor closed.

Let us now consider the subset A = {b, c, d} of X and determine the interior points, exterior points and boundary points of this subset A. Applying the above definitions we arrive at the following conclusions: The points c and d are each interior points of A since

c, d ε {c, d} A

where {c, d} is an open set. The point b ε A is not an interior point of A and thus int(A) = {c, d}. There is a single point exterior to A: the point a. It is exterior to A since it is interior to the complement Ac = {a, e} of A. The boundary of A then consists of the points b and c i.e. b(A) = {b, e}. Note that we have just rigorously applied the definitions above. There are no intuitive meanings associated with these interior, exterior and boundary points as there are when we are talking about points sets in continua. We simply crank them out from the definitions.

Example 2. Let X be a discrete topological space. Then every subset of X is open. Therefore every subset of X is also closed since its complement is open. In other words, all subsets of X are both open and closed.

Theorem 2. Let X be a topological space. Then the collection of closed subsets of X possess the following properties:

1) X and ∅ are closed sets.

2) The intersection of any number of closed sets is closed.

3) The union of any finite number of closed sets is closed.

Because of this theorem one could define a topology on a space using closed sets instead of open sets.

Theorem 3. Let A be a subset of topological space X. The closure of A is the union of the interior and boundary of A, i.e.

Theorem 4. A subset A of a topological space X is closed if and only if A contains each of its limit points.

Theorem 5. Let be the closure of set A. Then

1) is closed

2) if G is a closed superset of A, then

3) A is closed A =

Theorem 6. Let A be a subset of topological space X. Then the closure of A is the union of A and its set of limit points, i.e.

Properties of closure.

Def. Dense set. A subset A of a topological space X is said to be dense in X if the closure of A is X i.e. = X.

Def. Nowhere dense set. A subset A of a topological space X is said to be nowhere dense if the closure of A contains no interior points.

Example. Consider the subset A = {1, 1/2, 1/3, 1/4, .... } of R. This set has one limit point, 0. Thus = {0, 1, 1/2, 1/3, 1/4, .... }. has no interior points so A is nowhere dense in R.

Coarser and finer topologies. Suppose T1 and T2 are two topologies on a set X and that T1 is a subset of T2. Then we say that T1 is coarser than T2 — or that T2 is finer than T1. If T1 is a proper subset of T2 we say that T1 is strictly coarser than T2 — and that T2 is strictly finer than T1. Two topologies may not, of course, be comparable.

Example 3. Consider the discrete topology D, the indiscrete topology J, and any other topology T on any set X. Then J is coarser than T and T is coarser than D.

References

1. Lipschutz. General Topology

2. Simmons. Introduction to Topology and Modern Analysis

3. Munkres. Topology, A First Course

4. James & James. Mathematics Dictionary