In the branch of mathematics known as topology, the topologist's sine curve is an example that has several interesting properties. Mathematics is the study of quantity, structure, space and change. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...

It can be defined as a subset of the Euclidean plane as follows. Let S be the graph of the function sin(1/x) over the interval (0, 1]. Now let T be S union {(0,0)}. Give T the subset topology as a subset of the plane. T has the following properties:

• It is connected but not locally connected or path connected.

• It is not locally compact, but it is the continuous image of a locally compact space. (Namely, let V be the space {−1} union the interval (0, 1], and use the map f from V to T defined by f(−1) = (0, 0) and f(x) = (x, sin(1/x)).)

Two variations of the topologist's sine curve have other interesting properties. In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...

The closed topologist's sine curve can be defined by taking the same set S defined above, and adding to it the set {(0, y) | y is in the interval [−1, 1] }. It is closed, but has similar properties to the topologist's sine curve -- it too is connected but not locally connected or path-connected. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...

The extended topologist's sine curve can be defined by taking the topologist's sine curve and adding to it the set {(x, 1) | x is in the interval [0, 1] }. It is arc connected but not locally connected. In topology and related branches of mathematics, a topological space X is said to be connected if it cannot be divided into two disjoint nonempty open sets whose union is the entire space. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...

Image of the curve

This is a crude plot of the Topologist's sine curve. There are two important notes about this plot.

1. As x approaches zero, 1/x approaches infinity at an increasing rate. This is why the frequency of the sine wave appears to increase on the left side of the graph.

2. As x increases the curve asymptotically approaches zero.

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).