Topology is a mathematical field of exquisite beauty and refinement. It functions to abstract and generalize spatial relationship. For example, a sphere and a cube have the same topology, but a sphere and a torus have a different topology.

This material presents the rudiments of point-set topology. The other division of the subject is algebraic topology Point-set topology involves reducing the relationship among points to its most general level and formulating axioms in order to prove theorems of greatest generality. Another line of development in topology involves seeking special cases where theorems of greater definiteness can be proven.

A topology for a set S is defined as a collection of subsets of S, T={O_α: α ∈ I), such that

• The union of any collection of elements of T, finite or infinite, is a member of T.
• The intersection of any finite collection of elements of T is also a member of T.
• The whole set S and the null set ∅ are member of T.

There are many topologies which can be defined for a given set. In particular there are two extreme cases: the discrete topology and the indiscrete topology. The discrete topology for a set S is the collection of all subsets of S. The indiscrete topology for S is the collection consisting of only the whole set S and the null set ∅.

The sets in the topology T for a set S are defined as open. Thus openness is not a property determinable from the set itself; openness is a property of a set with respect to a topology. In the discrete topology any subset of S is open. In the discrete topology no subset of S other than S and ∅ are open.

Note that in any topology there are at least two sets which are both open and closed, S and ∅. In the discrete topology all subsets of S are both open and closed.

A set is defined as closed if its complement with respect to S is open; i.e., C is closed if the set of all elements of S that are not in C, (S-C), is open. The collection of all closed sets with respect to a topology T, T'={C_α: (S-C_α)∈T}, has the properties:

• The intersection of any collection of elements of T', finite or infinite, is a member of T'.
• The union of any finite collection of elements of T' is also a member of T'.
• The whole set S and the null set ∅ are member of T'.

The topology of a set could be defined in terms of its closed sets instead of its open sets. There is a duality involved if which (openness, unlimited union, finite intersection) corresponds to (closedness, finite union, unlimited intersection).

Lemma 1: If there is a collection of sets T'= {C_α} satisfying the above conditions then the collection of the complements of those sets T = {(S-C_α)} defines a topology for S.
 

Proof:
Let {(S-C_β)} be any subcollection of complements elements of T'. Consider ∪{(S-C_β)}. By de Morgan's Laws this is equal to S - ∩{C_β}. Since by assumption ∩{C_β} belongs to T' its complement belongs to T. On the other hand, consider {(S-C_γ)} is any finite collection of complements of elements of T'. The intersection of this set is by de Morgan's Law equal to S-∪{C_γ}. By assumption ∪{C_γ} belongs to T' and therefore its complement belongs to T. Finally, consider the sets S and ∅. S is a member of T' and therefore its complement ∅ is a member of T. The empty set ∅ is a member of T' and therefore its complement S is a member of T.

There are other equivalent ways to define a topology for a set besides open sets and closed sets. Kuratowski's closure operation is one of those other ways. Another way is in terms of the limit points of subsets of S.

An element p of X is a limit point of X with respect to the topology T of S if every open set of T containing p also contains an element of X distinct from p. With this definition a point p cannot be a limit point of the set consisting of p alone.

In the indiscrete topology all points are limit points of any subset X of S which inclues points other than because the only open set containing a point p is the whole S which necessarily contains points of X other than p. In the discrete topology no point is the limit point of any subset because for any point p the set {p} is open but does not contain any point of any subset X.

The closure of a set Q is the union of the set with its limit points. and it will denoted here as K(Q), since HTML does not have an overbar tag required for the usual notation. Kuratowski was able to define the topology of a set in terms of a closure operation which is a mapping of subsets of S into subsets of S (P(S)→P(S)), which has the following properties:

• The closure of the union of two sets is the union of their closures, K(X∪Y)=K(X)∪K(Y).
• The closure of the empty set ∅ is the empty set ∅, K(∅)=∅.
• The closure of a singleton set {p} is the singleton set {p} itself, K({p})={p}.
• The closure of the closure of a set is simply the closure of the set, K(K(X))=K(X).

The derivation of a topology for a set that has a Kuratowski closure operation is given in Appendix I.

The limit point concept can also be used as the basis for defining the topology of a set. Hocking and Young in their text Topology define topological space in terms of the concept of limit point and make it distinct from a pair (S,T) which is merely a set with a topology, a topologized set. Although topology can be defined in this way there is an awkwardness to this approach. The open set approach is clearer and less subject to confusion. Appendix II gives more information on the limit point approach to defining a topology for a set.

The collection of subsets of S in a topology may be excessive. It might be possible to define the collection from a more limited collection through the union of elements in this more limited collection. Such a collection is called a basis for a topology.

Let V={B_δ} be collection of subsets of S such that:

• ∪{B_δ}=S
• For any pair of elements of V, say B_δ1 and B_δ2 and any point p belonging to B_δ1∩B_δ2 there exists an element of V, say B_δ3, such that p∈B_δ3⊆(B_δ1∩B_δ2)

Such a collection V is a basis for a topology T if each element of T can be represented as the union of elements of V.

It can be shown that for any collection V satisfying the above conditions the union of its elements forms a topology. Thus basis collections are another way of defining topologies. .

Functions

An important concept in topology is that of a continuous function. In ordinary analysis continuity of a function f:D→R means that if x and y are close together in D then f(x) and f(y) are close together in R. In topology the situation is more complex. In particular, the continuity of a function is not a property of the rule for determining f(x) from x; it is a property of the topologies of the domain set and the range set. Thus in some topologies of the domain and range a given function may be continuous and in other it is not continuous.

The function notation is extended to apply to sets; i.e., f(X) is the set of elements {y: y=f(x) for some x∈X}.

A function f from the set S with topolgy T to the set R with the topology U is continuous if the set of points in S which map into an open set of U is open with respect to the topology T. In symbols this is expressed as

Definition of Continuity: f:S→R is continuous
with respect to the topology T of S and U of R
if and only if
for any open set W in R (W ∈ U)
f^-1(W) is open in T (f^-1(W) ∈ T)
 

The symbol f^-1(W) does not imply that the function has an inverse; it only refers to the set of points which map into the set W. In particular f^-1(W) may be the empty set ∅. If the topologies of the domain and range are the discrete topologies then any function is continuous. If they are the indiscrete topologies then any function is also continuous because the inverse image of the whole set of the range is the whole set of the domain of the function. It is very common in the literature of topology for authors to incorrectly treat continuity as if it were a property of the rule relating the domain set to the range set rather than being topologically dependent.

Connectedness

Another concept in which topological dependence is crucial is connectedness. A topological space is separated if it can be represented as the union of two disjoint, non-empty open sets. A topological space is connected if it is not separated. Disjointness and non-emptiness are set-theoretic but openness is topology dependent. This means that one cannot tell whether a topological space is connected or not by just looking at the set. In the indiscrete topology no set is separated because the only nonempty open set is the whole set. Therefore in the indiscrete topology all sets are connected. On the other hand, in the discrete topology no set with more than one point is connected. This is because any such set can be partitioned into two dispoint, nonempty subsets. Since in the discrete topology all subsets are open this partition constitutes a separation and hence the set is not connected.

The above material emphasizes the topology-dependent character of the concepts of openness, closedness, limit point, continuity and connectedness. It was important to establish this point before dealing with further topics in topology.

Coverings and Compactness

A collection of subsets of S, {S_delta;}, is said to cover a subset X if X is contained in the union of the collection; i.e., X ⊆ ∪{S_delta;}. If all of the sets in the collection are open sets then it is an open covering.

The compactness of a set with respect to a topology is one of the most import concepts in analysis. In ordinary Euclidean space if a set is closed and bounded then a continuous function will attain its maximum and its minimum within the set. This is a very important property and compactness is a generalization of the notion of closed and bounded to more general topological spaces. The generalization takes a surprising form. A subset X of S is compact if and only if any open covering of X, {O_delta;: δ∈I}, has a finite subcovering.

Topological Spaces and Subspaces

The definitions presented above are, strictly speaking, for topological spaces (S,T). In order to extend them to a subset X of S a topological space for X must be created. This is done by creating a topology W for X from the topology T={O_α} through intersection with X; i.e., W = {X∩O_α}.

Lemma 2: If T={O_α} is a topology for S then
W={X∩O_α} is a topology for X⊂S.

Proof:

Consider any collection from W, say {X∩O_β}. The union ∪{X∩O_β} is equal to X∩(∪{O_β}). But ∪{O_β} is element of T and hence its intersection with X is an element of W.
Likewise the intersection of any finite collection from W, say ∩{X∩O_γ} is equal to x∩{O_γ} and ∩{O_γ} necessarily belongs to T and hence its intersection with X belongs to W.
The nulll set ∅ belongs to W because ∅=X∩∅.
The subset X itself belongs to W because X=X∩S.

Thus propositions concerning a topological space can be applied to a subset through the creation of a corresponding topology for the subset, which is called a topological subspace.

Topology focuses on a special case of a function; i.e., a homeomorphism. A homeomorphism between a domain set S and a range set R is a function such that

• f:S→R is onto; i.e., for every point r in R there exist at least one s in S such that f(s)=r
• A point p of a subset X of S is a limit point of X if and only if f(p) is a limit point of f(X).

The essence of homeomorphism is that topological properties are preserved under homeomorphisms.

(To be continued.)

Appendix I: The Kuratowski Closure Operation

Let the function K:P(S)→P(S) be such that:

1. K(X∪Y) = K(X)∪K(Y)
2. K(∅) = ∅
3. K({p}) = {p}
4. K(K(X)) = K(X)

Lemma K0: X⊆K(X)

Proof: Consider any set X and any element x of X. Let X'=(X-{x}). Since X=X'∪{x}, K(X)=K(X')∪K({x})=K(X')∪{x}. Therefore x is an element of K(X) and hence X⊆K(X).

Lemma 1: If X ⊆ Y then K(X) ⊆ K(Y).

Proof: If X ⊆ Y then X∪Y=Y. Taking the closure, K(), of both sides of this set identity gives

K(X∪Y) = K(X)∪K(Y) = K(Y)
which implies that
K(X) ⊆ K(Y)
 

Consider the collection L = {X: X⊂S and K(X)=X}. It is to be shown that this collection constitutes a collection of closed sets in which arbitrary intersection, finite unions, the null set ∅, and the whole set S belong to the collection.

Lemma K2: If K(X_α)=X_α for α∈I then K(∩X_α) ⊆ ∩K(X_α)

Proof: For each β∈I, ∩X_α ⊆ X_β. Therefore ∩X_α ⊆ K(X_β) for each β and hence ∩X_α ⊆ ∩K(X_β)

Since K(X_β)=X_β this means that ∩X_α ⊆ ∩K(X_β). But by Lemma K0, ∩X_β ⊆ K(∩X_β). These two relationships imply that

K(∩X_β) = ∩X_β

and thus ∩X_β belongs to L.

Lemma 3: If K(X)=X and K(Y)=Y then K(X∪Y)=X∪Y.

Proof: K(X∪Y) = K(X)∪K(Y), but K(X) is the same as X and K(Y) is the same as Y so K(X∪Y)=X∪Y.

This means that if X and Y belong to L then X∪Y also belongs to L. This can be extended to any finite union.

Clearly ∅=K(∅) and S=K(S) belong to the collection.

Therefore a Kuratowski closure operation generates a collection of subsets of S which satisfy the closed set conditions and thus establish a topology for the set.

(To be continued.)