Continuity is conventionally associated with functions defined on the real line or higher-dimensional Euclidean spaces. Topology is concerned with the abstraction of continuity to maps between more general sets. The subject is vast and its study can take many forms depending on the nature of the structures considered. They include the areas of point-set, combinatorial, algebraic, and differential topology.
A topological space has a distinguished collection of subsets known as open sets. An open subset U of the real line is one for which every point of U is a subset of some real interval wholly contained in U. Thus, every point of U is the interior of U. Openness of a set U can also be expressed using the Euclidean distance, or metric, by saying that for every point all points within a sufficiently small distance of x also lie in U. Thus, metrics can be used to create open sets. Complements of open sets are said to be closed. The simplest examples of open and closed sets in are, respectively, the “open interval” (a, b) of all points between the numbers a and b excluding the end points, and the “closed interval” Also, sets can be neither open nor closed, for example
A topology on a set X is defined by its collection of open subsets, τ, which then makes X a topological space (X, τ). The collection of open subsets must include both X and the empty set any union of elements of τ, and the intersection of any finite collection of elements of τ. The conditions for a topology can equally well be cast in terms of closed sets. Also, a set can have many different topologies.
In any sophisticated mathematical structure, there is usually a way of relating two objects. For example, if we are only considering sets, say X and Y, we consider maps f :X→Y. The natural equivalence for sets X, Y would be the existence of a map f :X→Y which is both one-one and onto, that is, a bijection. When topologies are placed on X and Y, it is natural to consider maps f :X→Y that are continuous. The metric definition of continuity for maps or more generally can be shown to be equivalent to the topological definition: f :X→Y is continuous if V is an open subset of Y; the set f−1(V) is an open subset of X.
This alternative definition of continuity is the one that makes topology a key mathematical discipline of widespread importance.
The corresponding equivalence for topological spaces X and Y requires a map h:X→Y, which is both (i) a bijection, and (ii) bicontinuous; that is, both h and its inverse h−1 are continuous. Such a map is called a homeomorphism. The spaces X and Y are said to be topologically equivalent or homeomorphs. Any subset S of a topological space X can be made into a topological space by declaring the intersections of open sets of X with S to be open sets of S. In fact, this collection of subsets of S forms a topology on S, called the subspace topology. Thus, important geometrical objects which are subsets of Euclidean spaces such as the circle, the sphere (and therefore all classical polyhedra), the torus, the pretzel, and the Klein bottle (see Figure 1) can all be seen as topological spaces when endowed with the subspace topology.
An important property of continuous functions defined on the real numbers is that a continuous function defined on a bounded closed interval attains its bounds; that is, there exist points at which the function takes its maximum and minimum value. This is not true if the “closed, bounded” condition is relaxed. For example, f (x)=x is not bounded on the real line which is a closed (but not bounded) set. Also, f(x)=x does not attain its bounds on the bounded (but not closed) set (0, 1). The bounded closed interval on is called a compact set. Again, such a set can be defined solely in terms of open sets, and so we can define the concept of a compact topological space (Munkres, 2000).
Another key result in elementary analysis is that the continuous image of an interval of real numbers is also an interval. This property is often used to find roots of a continuous function by finding values for which f(a)·f(b)<0, the so-called “intermediate value” theorem. In the generalization of this result to topological spaces, the key property of the interval is its connectedness. The analogous result for topological spaces is that the continuous image of a connected set is also a connected set. The characterization of connectedness in the real numbers can be described purely in terms of the properties of the open sets on the real line, that is, in terms of the Euclidean topology on
Thus, both compactness and connectedness are topological properties in the sense that they can be described purely in terms of properties of the open sets of a topological space (Munkres, 2000).
In some areas of topology, the importance of the open sets is not so apparent and other features of the topological space are considered. For example, polyhedra such as the cube, tetrahedron, and dodecahedron are finite ways of building homeomorphic images of the standard sphere. We note that for all such constructs, the number of vertices (V), edges (E), and faces (F) satisfy the condition V−E+F=2, the Euler characteristic of the sphere. The torus, when built up in terms of faces, edges, and vertices, has the property that its Euler characteristic is zero. Given that the numbers V, E, and F are conserved by homeomorphism, we see that the torus and sphere having different Euler characteristics not only makes them look “different,” but ensures that they are not homeomorphic; that is, they are topologically distinct.
Note that not all spaces can be easily distinguished using topology. For example, the subsets of rational numbers, and the irrationals, I, of the real line are both topologically dense sets in that is, for both sets, the smallest closed superset is the whole interval Also, both are neither open nor closed. However, note that there is no bijection between the sets and
I because they have different cardinalities and, thus, cannot be homeomorphs.
The topological spaces mentioned above, such as the torus, sphere, and pretzel, can be easily visualized within the three-dimensional Euclidean space in which we locally live. However, it is not difficult to see how we can start to construct spaces in which are more geometrically demanding. The simplest is the Möbius strip, M, which is derived from the rectangle by pasting together one opposite pair of edges of a band with a half-twist, see Figure 1(d). Note that the Möbius band has only one “side”; just draw a pen line along the spine of the band and the line arrives on the opposite side of the paper from its initial point. Continuing the line brings a return to the initial point of the curve. By comparison, one cannot get from one side of a cylinder to the other without passing across one of its edges. The Möbius band is topologically different from the cylinder for several reasons. For instance, the boundary of the cylinder consists of two disjoint circles whereas that of the Möbius band is a single circle.
DAVID ARROWSMITH
Further Reading
Dieudonné, J. 1985. The beginnings of topology from 1850 to 1914. Proceedings of the Conference on Mathematical Logic 2(Siena, 1985), 585–600
Dieudonné, J. 1989. A History of Algebraic and Differential Topology, 1900–1960, Boston: Birkhäuser
Lefschetz, S. 1970. The early development of algebraic topology. Boletim da Sociedade Brasileira de Matematica, 1(1): 1–48
Mendelson, B. 1990. Introduction to Topology, New York: Dover
Munkres, J.R. 2000. Topology, Upper Saddle River, NJ: Prentice- Hall
Stillwell, J.C. 1993. Classical Topology and Combinatorial Group Theory, 2nd edition, New York: Springer
Weil, A. 1979. Riemann, Betti and the birth of topology. Archive for the History of Exact Sciences, 20(2): 91–96
is one for which every point of U is a subset of some real interval wholly contained in U. Thus, every point of U is the interior of U. Openness of a set U can also be expressed using the Euclidean distance, or metric, by saying that for every point 
all points within a sufficiently small distance of x also lie in U. Thus, metrics can be used to create open sets. Complements of open sets are said to be closed. The simplest examples of open and closed sets in 
are, respectively, the “open interval” (a, b) of all points between the numbers a and b excluding the end points, and the “closed interval” 
Also, sets can be neither open nor closed, for example 
any union of elements of τ, and the intersection of any finite collection of elements of τ. The conditions for a topology can equally well be cast in terms of closed sets. Also, a set can have many different topologies.
or more generally 
can be shown to be equivalent to the topological definition: f :X→Y is continuous if V is an open subset of Y; the set f−1(V) is an open subset of X.
which is a closed (but not bounded) set. Also, f(x)=x does not attain its bounds on the bounded (but not closed) set (0, 1). The bounded closed interval on 
is called a compact set. Again, such a set can be defined solely in terms of open sets, and so we can define the concept of a compact topological space (Munkres, 2000).
by finding values 
for which f(a)·f(b)<0, the so-called “intermediate value” theorem. In the generalization of this result to topological spaces, the key property of the interval is its connectedness. The analogous result for topological spaces is that the continuous image of a connected set is also a connected set. The characterization of connectedness in the real numbers can be described purely in terms of the properties of the open sets on the real line, that is, in terms of the Euclidean topology on 
and the irrationals, I, of the real line 
are both topologically dense sets in 
that is, for both sets, the smallest closed superset is the whole interval 
Also, both are neither open nor closed. However, note that there is no bijection between the sets 
and 
in which we locally live. However, it is not difficult to see how we can start to construct spaces in 
which are more geometrically demanding. The simplest is the Möbius strip, M, which is derived from the rectangle by pasting together one opposite pair of edges of a band with a half-twist, see 
